← Corresponding angles across an axis of symmetry · Transformations Preserve Measures

Corresponding angles across an axis of symmetry · 12 practice problems

4.G.A.34.MD.C.78.G.A.5

Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy answer: 145 degrees

The figure on the right is a line-symmetric figure. Find the measure, in degrees, of the marked angle.

The figure is symmetric about the horizontal line through the right-end vertex $E$ (its axis of symmetry). Going around the figure, the vertices are $S$ (top), $F$ , the right-end vertex $E$ , then $D$ , $C$ , $B$ along the bottom, and the left-end vertex $A$ . The marked angles measure $40^\circ$ at vertex $E$ , $120^\circ$ at the lower vertex $D$ , and $65^\circ$ at the lower-left vertex $B$ . The angle to find is the marked angle at the top vertex $S$ .

Show solution

1 · Understandwhat's really being asked

A figure with a horizontal axis of symmetry has three marked angles (40 at E, 120 at D, 65 at B). We need the angle at the top vertex S.

Givens

An exterior mark of 40 degrees at E, which lies on the axis.
The interior angle at D is 120 degrees.
An exterior mark of 65 degrees at B.

Unknowns

The marked angle at S.

Constraints

The axis reflects the top half onto the bottom half, so matching vertices have equal angles.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Cut the figure along the axis. The top half is a pentagon, and every angle in it except the one we want can be read off using symmetry. Then the pentagon's angle sum finishes it.

3 · Execute6 carry out the plan

1Draw the axis and split off the top pentagon

#1 Draw a Diagram 4.G.A.3

Draw the axis of symmetry through O and E. It divides the figure into a top half and a bottom half that are mirror images. The top half is the five-cornered figure O-A-S-F-E.

\text{top half} = \text{pentagon } O\text{-}A\text{-}S\text{-}F\text{-}E

One pentagon instead of a seven-sided figure.

2Find the angle at E using the 40 degree mark

#7 Identify Subproblems 4.MD.C.7

The 40 degree mark at E is between side FE and the straight extension of the bottom side, so the interior angle of the whole figure at E is 180 - 40 = 140 degrees. E sits on the axis, and the axis splits that angle into two equal halves, so the pentagon's angle at E is half of it.

180^{\circ} - 40^{\circ} = 140^{\circ}, \quad 140^{\circ} \div 2 = 70^{\circ}

70 degrees at E, inside the pentagon.

3Find the angle at A using the 65 degree mark

#1 Draw a Diagram 4.G.A.3

The 65 degree mark at B is outside the figure, so the interior angle at B is 180 - 65 = 115 degrees. Because A is the mirror image of B, the corresponding angle at A is the same 115 degrees.

180^{\circ} - 65^{\circ} = 115^{\circ}

115 degrees at A.

4Find the angle at F by corresponding angles

#1 Draw a Diagram 4.G.A.3

Vertex F at the top corresponds to vertex D at the bottom, whose interior angle is given as 120 degrees. So the angle at F is also 120 degrees.

\angle SFE = \angle CDE = 120^{\circ}

120 degrees at F.

5Use the right angle on the axis at O

#1 Draw a Diagram 4.G.A.3

The segment AB joins A to its mirror image B, so it crosses the axis at a right angle. That makes the angle A-O-E inside the top pentagon a right angle.

\angle AOE = 90^{\circ}

90 degrees at O, for free.

6Subtract from the pentagon's 540 degrees

#7 Identify Subproblems 8.G.A.5

The five angles of a five-sided figure always add to 540 degrees (it splits into three triangles, 3 x 180). The top pentagon's angles are 90 at O, 115 at A, 120 at F and 70 at E, so what is left over is the angle at S.

540^{\circ} - 90^{\circ} - 115^{\circ} - 120^{\circ} - 70^{\circ} = 145^{\circ}

The marked angle is 145 degrees.

Answer: 145 degrees

4 · Reviewdoes it hold up?

Add the pentagon back up: 90 + 115 + 145 + 120 + 70 = 540, which is the 540 a pentagon must have.

Another way: The whole seven-sided figure works too: its angles add to (7 - 2) x 180 = 900 degrees. Doubling the mirrored pairs gives 2 x (115 + 145 + 120) + 140 = 900 degrees -- the same 900.

Standardsmin grade 8

4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Copying angles from the bottom half to the top half.
4.MD.C.7 Recognize angle measure as additive and solve for unknown angles — Turning exterior marks into interior angles and halving at E.
8.G.A.5 Use informal arguments to establish facts about angles of triangles and polygons — The 540 degree angle sum of a pentagon.

💡Takeaway. Fold a symmetric figure along its axis and every angle you can see on one side tells you the matching one on the other.

Variant 2 easy answer: 140 degrees

The figure on the right is a line-symmetric figure. Find the measure, in degrees, of the marked angle.

The figure is symmetric about the horizontal line through the right-end vertex $E$ (its axis of symmetry). Going around the figure, the vertices are $S$ (top), $F$ , the right-end vertex $E$ , then $D$ , $C$ , $B$ along the bottom, and the left-end vertex $A$ . The marked angles measure $50^\circ$ at vertex $E$ , $125^\circ$ at the lower vertex $D$ , and $60^\circ$ at the lower-left vertex $B$ . The angle to find is the marked angle at the top vertex $S$ .

Show solution

1 · Understandwhat's really being asked

A figure with a horizontal axis of symmetry has three marked angles (50 at E, 125 at D, 60 at B). We need the angle at the top vertex S.

Givens

An exterior mark of 50 degrees at E, which lies on the axis.
The interior angle at D is 125 degrees.
An exterior mark of 60 degrees at B.

Unknowns

The marked angle at S.

Constraints

The axis reflects the top half onto the bottom half, so matching vertices have equal angles.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Cut the figure along the axis. The top half is a pentagon, and every angle in it except the one we want can be read off using symmetry. Then the pentagon's angle sum finishes it.

3 · Execute6 carry out the plan

1Draw the axis and split off the top pentagon

#1 Draw a Diagram 4.G.A.3

Draw the axis of symmetry through O and E. It divides the figure into a top half and a bottom half that are mirror images. The top half is the five-cornered figure O-A-S-F-E.

\text{top half} = \text{pentagon } O\text{-}A\text{-}S\text{-}F\text{-}E

One pentagon instead of a seven-sided figure.

2Find the angle at E using the 50 degree mark

#7 Identify Subproblems 4.MD.C.7

The 50 degree mark at E is between side FE and the straight extension of the bottom side, so the interior angle of the whole figure at E is 180 - 50 = 130 degrees. E sits on the axis, and the axis splits that angle into two equal halves, so the pentagon's angle at E is half of it.

180^{\circ} - 50^{\circ} = 130^{\circ}, \quad 130^{\circ} \div 2 = 65^{\circ}

65 degrees at E, inside the pentagon.

3Find the angle at A using the 60 degree mark

#1 Draw a Diagram 4.G.A.3

The 60 degree mark at B is outside the figure, so the interior angle at B is 180 - 60 = 120 degrees. Because A is the mirror image of B, the corresponding angle at A is the same 120 degrees.

180^{\circ} - 60^{\circ} = 120^{\circ}

120 degrees at A.

4Find the angle at F by corresponding angles

#1 Draw a Diagram 4.G.A.3

Vertex F at the top corresponds to vertex D at the bottom, whose interior angle is given as 125 degrees. So the angle at F is also 125 degrees.

\angle SFE = \angle CDE = 125^{\circ}

125 degrees at F.

5Use the right angle on the axis at O

#1 Draw a Diagram 4.G.A.3

The segment AB joins A to its mirror image B, so it crosses the axis at a right angle. That makes the angle A-O-E inside the top pentagon a right angle.

\angle AOE = 90^{\circ}

90 degrees at O, for free.

6Subtract from the pentagon's 540 degrees

#7 Identify Subproblems 8.G.A.5

The five angles of a five-sided figure always add to 540 degrees (it splits into three triangles, 3 x 180). The top pentagon's angles are 90 at O, 120 at A, 125 at F and 65 at E, so what is left over is the angle at S.

540^{\circ} - 90^{\circ} - 120^{\circ} - 125^{\circ} - 65^{\circ} = 140^{\circ}

The marked angle is 140 degrees.

Answer: 140 degrees

4 · Reviewdoes it hold up?

Add the pentagon back up: 90 + 120 + 140 + 125 + 65 = 540, which is the 540 a pentagon must have.

Another way: The whole seven-sided figure works too: its angles add to (7 - 2) x 180 = 900 degrees. Doubling the mirrored pairs gives 2 x (120 + 140 + 125) + 130 = 900 degrees -- the same 900.

Standardsmin grade 8

4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Copying angles from the bottom half to the top half.
4.MD.C.7 Recognize angle measure as additive and solve for unknown angles — Turning exterior marks into interior angles and halving at E.
8.G.A.5 Use informal arguments to establish facts about angles of triangles and polygons — The 540 degree angle sum of a pentagon.

💡Takeaway. Fold a symmetric figure along its axis and every angle you can see on one side tells you the matching one on the other.

Variant 3 easy answer: 130 degrees

The figure on the right is a line-symmetric figure. Find the measure, in degrees, of the marked angle.

The figure is symmetric about the horizontal line through the right-end vertex $E$ (its axis of symmetry). Going around the figure, the vertices are $S$ (top), $F$ , the right-end vertex $E$ , then $D$ , $C$ , $B$ along the bottom, and the left-end vertex $A$ . The marked angles measure $60^\circ$ at vertex $E$ , $130^\circ$ at the lower vertex $D$ , and $50^\circ$ at the lower-left vertex $B$ . The angle to find is the marked angle at the top vertex $S$ .

Show solution

1 · Understandwhat's really being asked

A figure with a horizontal axis of symmetry has three marked angles (60 at E, 130 at D, 50 at B). We need the angle at the top vertex S.

Givens

An exterior mark of 60 degrees at E, which lies on the axis.
The interior angle at D is 130 degrees.
An exterior mark of 50 degrees at B.

Unknowns

The marked angle at S.

Constraints

The axis reflects the top half onto the bottom half, so matching vertices have equal angles.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Cut the figure along the axis. The top half is a pentagon, and every angle in it except the one we want can be read off using symmetry. Then the pentagon's angle sum finishes it.

3 · Execute6 carry out the plan

1Draw the axis and split off the top pentagon

#1 Draw a Diagram 4.G.A.3

Draw the axis of symmetry through O and E. It divides the figure into a top half and a bottom half that are mirror images. The top half is the five-cornered figure O-A-S-F-E.

\text{top half} = \text{pentagon } O\text{-}A\text{-}S\text{-}F\text{-}E

One pentagon instead of a seven-sided figure.

2Find the angle at E using the 60 degree mark

#7 Identify Subproblems 4.MD.C.7

The 60 degree mark at E is between side FE and the straight extension of the bottom side, so the interior angle of the whole figure at E is 180 - 60 = 120 degrees. E sits on the axis, and the axis splits that angle into two equal halves, so the pentagon's angle at E is half of it.

180^{\circ} - 60^{\circ} = 120^{\circ}, \quad 120^{\circ} \div 2 = 60^{\circ}

60 degrees at E, inside the pentagon.

3Find the angle at A using the 50 degree mark

#1 Draw a Diagram 4.G.A.3

The 50 degree mark at B is outside the figure, so the interior angle at B is 180 - 50 = 130 degrees. Because A is the mirror image of B, the corresponding angle at A is the same 130 degrees.

180^{\circ} - 50^{\circ} = 130^{\circ}

130 degrees at A.

4Find the angle at F by corresponding angles

#1 Draw a Diagram 4.G.A.3

Vertex F at the top corresponds to vertex D at the bottom, whose interior angle is given as 130 degrees. So the angle at F is also 130 degrees.

\angle SFE = \angle CDE = 130^{\circ}

130 degrees at F.

5Use the right angle on the axis at O

#1 Draw a Diagram 4.G.A.3

The segment AB joins A to its mirror image B, so it crosses the axis at a right angle. That makes the angle A-O-E inside the top pentagon a right angle.

\angle AOE = 90^{\circ}

90 degrees at O, for free.

6Subtract from the pentagon's 540 degrees

#7 Identify Subproblems 8.G.A.5

The five angles of a five-sided figure always add to 540 degrees (it splits into three triangles, 3 x 180). The top pentagon's angles are 90 at O, 130 at A, 130 at F and 60 at E, so what is left over is the angle at S.

540^{\circ} - 90^{\circ} - 130^{\circ} - 130^{\circ} - 60^{\circ} = 130^{\circ}

The marked angle is 130 degrees.

Answer: 130 degrees

4 · Reviewdoes it hold up?

Add the pentagon back up: 90 + 130 + 130 + 130 + 60 = 540, which is the 540 a pentagon must have.

Another way: The whole seven-sided figure works too: its angles add to (7 - 2) x 180 = 900 degrees. Doubling the mirrored pairs gives 2 x (130 + 130 + 130) + 120 = 900 degrees -- the same 900.

Standardsmin grade 8

4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Copying angles from the bottom half to the top half.
4.MD.C.7 Recognize angle measure as additive and solve for unknown angles — Turning exterior marks into interior angles and halving at E.
8.G.A.5 Use informal arguments to establish facts about angles of triangles and polygons — The 540 degree angle sum of a pentagon.

💡Takeaway. Fold a symmetric figure along its axis and every angle you can see on one side tells you the matching one on the other.

Variant 4 easy answer: 132 degrees

The figure on the right is a line-symmetric figure. Find the measure, in degrees, of the marked angle.

The figure is symmetric about the horizontal line through the right-end vertex $E$ (its axis of symmetry). Going around the figure, the vertices are $S$ (top), $F$ , the right-end vertex $E$ , then $D$ , $C$ , $B$ along the bottom, and the left-end vertex $A$ . The marked angles measure $84^\circ$ at vertex $E$ , $132^\circ$ at the lower vertex $D$ , and $42^\circ$ at the lower-left vertex $B$ . The angle to find is the marked angle at the top vertex $S$ .

Show solution

1 · Understandwhat's really being asked

A figure with a horizontal axis of symmetry has three marked angles (84 at E, 132 at D, 42 at B). We need the angle at the top vertex S.

Givens

An exterior mark of 84 degrees at E, which lies on the axis.
The interior angle at D is 132 degrees.
An exterior mark of 42 degrees at B.

Unknowns

The marked angle at S.

Constraints

The axis reflects the top half onto the bottom half, so matching vertices have equal angles.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Cut the figure along the axis. The top half is a pentagon, and every angle in it except the one we want can be read off using symmetry. Then the pentagon's angle sum finishes it.

3 · Execute6 carry out the plan

1Draw the axis and split off the top pentagon

#1 Draw a Diagram 4.G.A.3

Draw the axis of symmetry through O and E. It divides the figure into a top half and a bottom half that are mirror images. The top half is the five-cornered figure O-A-S-F-E.

\text{top half} = \text{pentagon } O\text{-}A\text{-}S\text{-}F\text{-}E

One pentagon instead of a seven-sided figure.

2Find the angle at E using the 84 degree mark

#7 Identify Subproblems 4.MD.C.7

The 84 degree mark at E is between side FE and the straight extension of the bottom side, so the interior angle of the whole figure at E is 180 - 84 = 96 degrees. E sits on the axis, and the axis splits that angle into two equal halves, so the pentagon's angle at E is half of it.

180^{\circ} - 84^{\circ} = 96^{\circ}, \quad 96^{\circ} \div 2 = 48^{\circ}

48 degrees at E, inside the pentagon.

3Find the angle at A using the 42 degree mark

#1 Draw a Diagram 4.G.A.3

The 42 degree mark at B is outside the figure, so the interior angle at B is 180 - 42 = 138 degrees. Because A is the mirror image of B, the corresponding angle at A is the same 138 degrees.

180^{\circ} - 42^{\circ} = 138^{\circ}

138 degrees at A.

4Find the angle at F by corresponding angles

#1 Draw a Diagram 4.G.A.3

Vertex F at the top corresponds to vertex D at the bottom, whose interior angle is given as 132 degrees. So the angle at F is also 132 degrees.

\angle SFE = \angle CDE = 132^{\circ}

132 degrees at F.

5Use the right angle on the axis at O

#1 Draw a Diagram 4.G.A.3

The segment AB joins A to its mirror image B, so it crosses the axis at a right angle. That makes the angle A-O-E inside the top pentagon a right angle.

\angle AOE = 90^{\circ}

90 degrees at O, for free.

6Subtract from the pentagon's 540 degrees

#7 Identify Subproblems 8.G.A.5

The five angles of a five-sided figure always add to 540 degrees (it splits into three triangles, 3 x 180). The top pentagon's angles are 90 at O, 138 at A, 132 at F and 48 at E, so what is left over is the angle at S.

540^{\circ} - 90^{\circ} - 138^{\circ} - 132^{\circ} - 48^{\circ} = 132^{\circ}

The marked angle is 132 degrees.

Answer: 132 degrees

4 · Reviewdoes it hold up?

Add the pentagon back up: 90 + 138 + 132 + 132 + 48 = 540, which is the 540 a pentagon must have.

Another way: The whole seven-sided figure works too: its angles add to (7 - 2) x 180 = 900 degrees. Doubling the mirrored pairs gives 2 x (138 + 132 + 132) + 96 = 900 degrees -- the same 900.

Standardsmin grade 8

4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Copying angles from the bottom half to the top half.
4.MD.C.7 Recognize angle measure as additive and solve for unknown angles — Turning exterior marks into interior angles and halving at E.
8.G.A.5 Use informal arguments to establish facts about angles of triangles and polygons — The 540 degree angle sum of a pentagon.

💡Takeaway. Fold a symmetric figure along its axis and every angle you can see on one side tells you the matching one on the other.

Variant 5 medium answer: 135 degrees

The figure on the right is a line-symmetric figure. Find the measure, in degrees, of the marked angle.

The figure is symmetric about the horizontal line through the right-end vertex $E$ (its axis of symmetry). Going around the figure, the vertices are $S$ (top), $F$ , the right-end vertex $E$ , then $D$ , $C$ , $B$ along the bottom, and the left-end vertex $A$ . The marked angles measure $70^\circ$ at vertex $E$ , $135^\circ$ at the lower vertex $D$ , and $55^\circ$ at the lower-left vertex $B$ . The angle to find is the marked angle at the top vertex $S$ .

Show solution

1 · Understandwhat's really being asked

A figure with a horizontal axis of symmetry has three marked angles (70 at E, 135 at D, 55 at B). We need the angle at the top vertex S.

Givens

An exterior mark of 70 degrees at E, which lies on the axis.
The interior angle at D is 135 degrees.
An exterior mark of 55 degrees at B.

Unknowns

The marked angle at S.

Constraints

The axis reflects the top half onto the bottom half, so matching vertices have equal angles.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Cut the figure along the axis. The top half is a pentagon, and every angle in it except the one we want can be read off using symmetry. Then the pentagon's angle sum finishes it.

3 · Execute6 carry out the plan

1Draw the axis and split off the top pentagon

#1 Draw a Diagram 4.G.A.3

Draw the axis of symmetry through O and E. It divides the figure into a top half and a bottom half that are mirror images. The top half is the five-cornered figure O-A-S-F-E.

\text{top half} = \text{pentagon } O\text{-}A\text{-}S\text{-}F\text{-}E

One pentagon instead of a seven-sided figure.

2Find the angle at E using the 70 degree mark

#7 Identify Subproblems 4.MD.C.7

The 70 degree mark at E is between side FE and the straight extension of the bottom side, so the interior angle of the whole figure at E is 180 - 70 = 110 degrees. E sits on the axis, and the axis splits that angle into two equal halves, so the pentagon's angle at E is half of it.

180^{\circ} - 70^{\circ} = 110^{\circ}, \quad 110^{\circ} \div 2 = 55^{\circ}

55 degrees at E, inside the pentagon.

3Find the angle at A using the 55 degree mark

#1 Draw a Diagram 4.G.A.3

The 55 degree mark at B is outside the figure, so the interior angle at B is 180 - 55 = 125 degrees. Because A is the mirror image of B, the corresponding angle at A is the same 125 degrees.

180^{\circ} - 55^{\circ} = 125^{\circ}

125 degrees at A.

4Find the angle at F by corresponding angles

#1 Draw a Diagram 4.G.A.3

Vertex F at the top corresponds to vertex D at the bottom, whose interior angle is given as 135 degrees. So the angle at F is also 135 degrees.

\angle SFE = \angle CDE = 135^{\circ}

135 degrees at F.

5Use the right angle on the axis at O

#1 Draw a Diagram 4.G.A.3

The segment AB joins A to its mirror image B, so it crosses the axis at a right angle. That makes the angle A-O-E inside the top pentagon a right angle.

\angle AOE = 90^{\circ}

90 degrees at O, for free.

6Subtract from the pentagon's 540 degrees

#7 Identify Subproblems 8.G.A.5

The five angles of a five-sided figure always add to 540 degrees (it splits into three triangles, 3 x 180). The top pentagon's angles are 90 at O, 125 at A, 135 at F and 55 at E, so what is left over is the angle at S.

540^{\circ} - 90^{\circ} - 125^{\circ} - 135^{\circ} - 55^{\circ} = 135^{\circ}

The marked angle is 135 degrees.

Answer: 135 degrees

4 · Reviewdoes it hold up?

Add the pentagon back up: 90 + 125 + 135 + 135 + 55 = 540, which is the 540 a pentagon must have.

Another way: The whole seven-sided figure works too: its angles add to (7 - 2) x 180 = 900 degrees. Doubling the mirrored pairs gives 2 x (125 + 135 + 135) + 110 = 900 degrees -- the same 900.

Standardsmin grade 8

4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Copying angles from the bottom half to the top half.
4.MD.C.7 Recognize angle measure as additive and solve for unknown angles — Turning exterior marks into interior angles and halving at E.
8.G.A.5 Use informal arguments to establish facts about angles of triangles and polygons — The 540 degree angle sum of a pentagon.

💡Takeaway. Fold a symmetric figure along its axis and every angle you can see on one side tells you the matching one on the other.

Variant 6 medium answer: 123 degrees

The figure on the right is a line-symmetric figure. Find the measure, in degrees, of the marked angle.

The figure is symmetric about the horizontal line through the right-end vertex $E$ (its axis of symmetry). Going around the figure, the vertices are $S$ (top), $F$ , the right-end vertex $E$ , then $D$ , $C$ , $B$ along the bottom, and the left-end vertex $A$ . The marked angles measure $66^\circ$ at vertex $E$ , $138^\circ$ at the lower vertex $D$ , and $48^\circ$ at the lower-left vertex $B$ . The angle to find is the marked angle at the top vertex $S$ .

Show solution

1 · Understandwhat's really being asked

A figure with a horizontal axis of symmetry has three marked angles (66 at E, 138 at D, 48 at B). We need the angle at the top vertex S.

Givens

An exterior mark of 66 degrees at E, which lies on the axis.
The interior angle at D is 138 degrees.
An exterior mark of 48 degrees at B.

Unknowns

The marked angle at S.

Constraints

The axis reflects the top half onto the bottom half, so matching vertices have equal angles.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Cut the figure along the axis. The top half is a pentagon, and every angle in it except the one we want can be read off using symmetry. Then the pentagon's angle sum finishes it.

3 · Execute6 carry out the plan

1Draw the axis and split off the top pentagon

#1 Draw a Diagram 4.G.A.3

Draw the axis of symmetry through O and E. It divides the figure into a top half and a bottom half that are mirror images. The top half is the five-cornered figure O-A-S-F-E.

\text{top half} = \text{pentagon } O\text{-}A\text{-}S\text{-}F\text{-}E

One pentagon instead of a seven-sided figure.

2Find the angle at E using the 66 degree mark

#7 Identify Subproblems 4.MD.C.7

The 66 degree mark at E is between side FE and the straight extension of the bottom side, so the interior angle of the whole figure at E is 180 - 66 = 114 degrees. E sits on the axis, and the axis splits that angle into two equal halves, so the pentagon's angle at E is half of it.

180^{\circ} - 66^{\circ} = 114^{\circ}, \quad 114^{\circ} \div 2 = 57^{\circ}

57 degrees at E, inside the pentagon.

3Find the angle at A using the 48 degree mark

#1 Draw a Diagram 4.G.A.3

The 48 degree mark at B is outside the figure, so the interior angle at B is 180 - 48 = 132 degrees. Because A is the mirror image of B, the corresponding angle at A is the same 132 degrees.

180^{\circ} - 48^{\circ} = 132^{\circ}

132 degrees at A.

4Find the angle at F by corresponding angles

#1 Draw a Diagram 4.G.A.3

Vertex F at the top corresponds to vertex D at the bottom, whose interior angle is given as 138 degrees. So the angle at F is also 138 degrees.

\angle SFE = \angle CDE = 138^{\circ}

138 degrees at F.

5Use the right angle on the axis at O

#1 Draw a Diagram 4.G.A.3

The segment AB joins A to its mirror image B, so it crosses the axis at a right angle. That makes the angle A-O-E inside the top pentagon a right angle.

\angle AOE = 90^{\circ}

90 degrees at O, for free.

6Subtract from the pentagon's 540 degrees

#7 Identify Subproblems 8.G.A.5

The five angles of a five-sided figure always add to 540 degrees (it splits into three triangles, 3 x 180). The top pentagon's angles are 90 at O, 132 at A, 138 at F and 57 at E, so what is left over is the angle at S.

540^{\circ} - 90^{\circ} - 132^{\circ} - 138^{\circ} - 57^{\circ} = 123^{\circ}

The marked angle is 123 degrees.

Answer: 123 degrees

4 · Reviewdoes it hold up?

Add the pentagon back up: 90 + 132 + 123 + 138 + 57 = 540, which is the 540 a pentagon must have.

Another way: The whole seven-sided figure works too: its angles add to (7 - 2) x 180 = 900 degrees. Doubling the mirrored pairs gives 2 x (132 + 123 + 138) + 114 = 900 degrees -- the same 900.

Standardsmin grade 8

4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Copying angles from the bottom half to the top half.
4.MD.C.7 Recognize angle measure as additive and solve for unknown angles — Turning exterior marks into interior angles and halving at E.
8.G.A.5 Use informal arguments to establish facts about angles of triangles and polygons — The 540 degree angle sum of a pentagon.

💡Takeaway. Fold a symmetric figure along its axis and every angle you can see on one side tells you the matching one on the other.

Variant 7 medium answer: 125 degrees

The figure on the right is a line-symmetric figure. Find the measure, in degrees, of the marked angle.

The figure is symmetric about the horizontal line through the right-end vertex $E$ (its axis of symmetry). Going around the figure, the vertices are $S$ (top), $F$ , the right-end vertex $E$ , then $D$ , $C$ , $B$ along the bottom, and the left-end vertex $A$ . The marked angles measure $80^\circ$ at vertex $E$ , $140^\circ$ at the lower vertex $D$ , and $45^\circ$ at the lower-left vertex $B$ . The angle to find is the marked angle at the top vertex $S$ .

Show solution

1 · Understandwhat's really being asked

A figure with a horizontal axis of symmetry has three marked angles (80 at E, 140 at D, 45 at B). We need the angle at the top vertex S.

Givens

An exterior mark of 80 degrees at E, which lies on the axis.
The interior angle at D is 140 degrees.
An exterior mark of 45 degrees at B.

Unknowns

The marked angle at S.

Constraints

The axis reflects the top half onto the bottom half, so matching vertices have equal angles.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Cut the figure along the axis. The top half is a pentagon, and every angle in it except the one we want can be read off using symmetry. Then the pentagon's angle sum finishes it.

3 · Execute6 carry out the plan

1Draw the axis and split off the top pentagon

#1 Draw a Diagram 4.G.A.3

Draw the axis of symmetry through O and E. It divides the figure into a top half and a bottom half that are mirror images. The top half is the five-cornered figure O-A-S-F-E.

\text{top half} = \text{pentagon } O\text{-}A\text{-}S\text{-}F\text{-}E

One pentagon instead of a seven-sided figure.

2Find the angle at E using the 80 degree mark

#7 Identify Subproblems 4.MD.C.7

The 80 degree mark at E is between side FE and the straight extension of the bottom side, so the interior angle of the whole figure at E is 180 - 80 = 100 degrees. E sits on the axis, and the axis splits that angle into two equal halves, so the pentagon's angle at E is half of it.

180^{\circ} - 80^{\circ} = 100^{\circ}, \quad 100^{\circ} \div 2 = 50^{\circ}

50 degrees at E, inside the pentagon.

3Find the angle at A using the 45 degree mark

#1 Draw a Diagram 4.G.A.3

The 45 degree mark at B is outside the figure, so the interior angle at B is 180 - 45 = 135 degrees. Because A is the mirror image of B, the corresponding angle at A is the same 135 degrees.

180^{\circ} - 45^{\circ} = 135^{\circ}

135 degrees at A.

4Find the angle at F by corresponding angles

#1 Draw a Diagram 4.G.A.3

Vertex F at the top corresponds to vertex D at the bottom, whose interior angle is given as 140 degrees. So the angle at F is also 140 degrees.

\angle SFE = \angle CDE = 140^{\circ}

140 degrees at F.

5Use the right angle on the axis at O

#1 Draw a Diagram 4.G.A.3

The segment AB joins A to its mirror image B, so it crosses the axis at a right angle. That makes the angle A-O-E inside the top pentagon a right angle.

\angle AOE = 90^{\circ}

90 degrees at O, for free.

6Subtract from the pentagon's 540 degrees

#7 Identify Subproblems 8.G.A.5

The five angles of a five-sided figure always add to 540 degrees (it splits into three triangles, 3 x 180). The top pentagon's angles are 90 at O, 135 at A, 140 at F and 50 at E, so what is left over is the angle at S.

540^{\circ} - 90^{\circ} - 135^{\circ} - 140^{\circ} - 50^{\circ} = 125^{\circ}

The marked angle is 125 degrees.

Answer: 125 degrees

4 · Reviewdoes it hold up?

Add the pentagon back up: 90 + 135 + 125 + 140 + 50 = 540, which is the 540 a pentagon must have.

Another way: The whole seven-sided figure works too: its angles add to (7 - 2) x 180 = 900 degrees. Doubling the mirrored pairs gives 2 x (135 + 125 + 140) + 100 = 900 degrees -- the same 900.

Standardsmin grade 8

4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Copying angles from the bottom half to the top half.
4.MD.C.7 Recognize angle measure as additive and solve for unknown angles — Turning exterior marks into interior angles and halving at E.
8.G.A.5 Use informal arguments to establish facts about angles of triangles and polygons — The 540 degree angle sum of a pentagon.

💡Takeaway. Fold a symmetric figure along its axis and every angle you can see on one side tells you the matching one on the other.

Variant 8 medium answer: 120 degrees

The figure on the right is a line-symmetric figure. Find the measure, in degrees, of the marked angle.

The figure is symmetric about the horizontal line through the right-end vertex $E$ (its axis of symmetry). Going around the figure, the vertices are $S$ (top), $F$ , the right-end vertex $E$ , then $D$ , $C$ , $B$ along the bottom, and the left-end vertex $A$ . The marked angles measure $100^\circ$ at vertex $E$ , $145^\circ$ at the lower vertex $D$ , and $35^\circ$ at the lower-left vertex $B$ . The angle to find is the marked angle at the top vertex $S$ .

Show solution

1 · Understandwhat's really being asked

A figure with a horizontal axis of symmetry has three marked angles (100 at E, 145 at D, 35 at B). We need the angle at the top vertex S.

Givens

An exterior mark of 100 degrees at E, which lies on the axis.
The interior angle at D is 145 degrees.
An exterior mark of 35 degrees at B.

Unknowns

The marked angle at S.

Constraints

The axis reflects the top half onto the bottom half, so matching vertices have equal angles.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Cut the figure along the axis. The top half is a pentagon, and every angle in it except the one we want can be read off using symmetry. Then the pentagon's angle sum finishes it.

3 · Execute6 carry out the plan

1Draw the axis and split off the top pentagon

#1 Draw a Diagram 4.G.A.3

Draw the axis of symmetry through O and E. It divides the figure into a top half and a bottom half that are mirror images. The top half is the five-cornered figure O-A-S-F-E.

\text{top half} = \text{pentagon } O\text{-}A\text{-}S\text{-}F\text{-}E

One pentagon instead of a seven-sided figure.

2Find the angle at E using the 100 degree mark

#7 Identify Subproblems 4.MD.C.7

The 100 degree mark at E is between side FE and the straight extension of the bottom side, so the interior angle of the whole figure at E is 180 - 100 = 80 degrees. E sits on the axis, and the axis splits that angle into two equal halves, so the pentagon's angle at E is half of it.

180^{\circ} - 100^{\circ} = 80^{\circ}, \quad 80^{\circ} \div 2 = 40^{\circ}

40 degrees at E, inside the pentagon.

3Find the angle at A using the 35 degree mark

#1 Draw a Diagram 4.G.A.3

The 35 degree mark at B is outside the figure, so the interior angle at B is 180 - 35 = 145 degrees. Because A is the mirror image of B, the corresponding angle at A is the same 145 degrees.

180^{\circ} - 35^{\circ} = 145^{\circ}

145 degrees at A.

4Find the angle at F by corresponding angles

#1 Draw a Diagram 4.G.A.3

Vertex F at the top corresponds to vertex D at the bottom, whose interior angle is given as 145 degrees. So the angle at F is also 145 degrees.

\angle SFE = \angle CDE = 145^{\circ}

145 degrees at F.

5Use the right angle on the axis at O

#1 Draw a Diagram 4.G.A.3

The segment AB joins A to its mirror image B, so it crosses the axis at a right angle. That makes the angle A-O-E inside the top pentagon a right angle.

\angle AOE = 90^{\circ}

90 degrees at O, for free.

6Subtract from the pentagon's 540 degrees

#7 Identify Subproblems 8.G.A.5

The five angles of a five-sided figure always add to 540 degrees (it splits into three triangles, 3 x 180). The top pentagon's angles are 90 at O, 145 at A, 145 at F and 40 at E, so what is left over is the angle at S.

540^{\circ} - 90^{\circ} - 145^{\circ} - 145^{\circ} - 40^{\circ} = 120^{\circ}

The marked angle is 120 degrees.

Answer: 120 degrees

4 · Reviewdoes it hold up?

Add the pentagon back up: 90 + 145 + 120 + 145 + 40 = 540, which is the 540 a pentagon must have.

Another way: The whole seven-sided figure works too: its angles add to (7 - 2) x 180 = 900 degrees. Doubling the mirrored pairs gives 2 x (145 + 120 + 145) + 80 = 900 degrees -- the same 900.

Standardsmin grade 8

4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Copying angles from the bottom half to the top half.
4.MD.C.7 Recognize angle measure as additive and solve for unknown angles — Turning exterior marks into interior angles and halving at E.
8.G.A.5 Use informal arguments to establish facts about angles of triangles and polygons — The 540 degree angle sum of a pentagon.

💡Takeaway. Fold a symmetric figure along its axis and every angle you can see on one side tells you the matching one on the other.

Variant 9 hard answer: 118 degrees

The figure on the right is a line-symmetric figure. Find the measure, in degrees, of the marked angle.

The figure is symmetric about the horizontal line through the right-end vertex $E$ (its axis of symmetry). Going around the figure, the vertices are $S$ (top), $F$ , the right-end vertex $E$ , then $D$ , $C$ , $B$ along the bottom, and the left-end vertex $A$ . The marked angles measure $96^\circ$ at vertex $E$ , $148^\circ$ at the lower vertex $D$ , and $38^\circ$ at the lower-left vertex $B$ . The angle to find is the marked angle at the top vertex $S$ .

Show solution

1 · Understandwhat's really being asked

A figure with a horizontal axis of symmetry has three marked angles (96 at E, 148 at D, 38 at B). We need the angle at the top vertex S.

Givens

An exterior mark of 96 degrees at E, which lies on the axis.
The interior angle at D is 148 degrees.
An exterior mark of 38 degrees at B.

Unknowns

The marked angle at S.

Constraints

The axis reflects the top half onto the bottom half, so matching vertices have equal angles.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Cut the figure along the axis. The top half is a pentagon, and every angle in it except the one we want can be read off using symmetry. Then the pentagon's angle sum finishes it.

3 · Execute6 carry out the plan

1Draw the axis and split off the top pentagon

#1 Draw a Diagram 4.G.A.3

Draw the axis of symmetry through O and E. It divides the figure into a top half and a bottom half that are mirror images. The top half is the five-cornered figure O-A-S-F-E.

\text{top half} = \text{pentagon } O\text{-}A\text{-}S\text{-}F\text{-}E

One pentagon instead of a seven-sided figure.

2Find the angle at E using the 96 degree mark

#7 Identify Subproblems 4.MD.C.7

The 96 degree mark at E is between side FE and the straight extension of the bottom side, so the interior angle of the whole figure at E is 180 - 96 = 84 degrees. E sits on the axis, and the axis splits that angle into two equal halves, so the pentagon's angle at E is half of it.

180^{\circ} - 96^{\circ} = 84^{\circ}, \quad 84^{\circ} \div 2 = 42^{\circ}

42 degrees at E, inside the pentagon.

3Find the angle at A using the 38 degree mark

#1 Draw a Diagram 4.G.A.3

The 38 degree mark at B is outside the figure, so the interior angle at B is 180 - 38 = 142 degrees. Because A is the mirror image of B, the corresponding angle at A is the same 142 degrees.

180^{\circ} - 38^{\circ} = 142^{\circ}

142 degrees at A.

4Find the angle at F by corresponding angles

#1 Draw a Diagram 4.G.A.3

Vertex F at the top corresponds to vertex D at the bottom, whose interior angle is given as 148 degrees. So the angle at F is also 148 degrees.

\angle SFE = \angle CDE = 148^{\circ}

148 degrees at F.

5Use the right angle on the axis at O

#1 Draw a Diagram 4.G.A.3

The segment AB joins A to its mirror image B, so it crosses the axis at a right angle. That makes the angle A-O-E inside the top pentagon a right angle.

\angle AOE = 90^{\circ}

90 degrees at O, for free.

6Subtract from the pentagon's 540 degrees

#7 Identify Subproblems 8.G.A.5

The five angles of a five-sided figure always add to 540 degrees (it splits into three triangles, 3 x 180). The top pentagon's angles are 90 at O, 142 at A, 148 at F and 42 at E, so what is left over is the angle at S.

540^{\circ} - 90^{\circ} - 142^{\circ} - 148^{\circ} - 42^{\circ} = 118^{\circ}

The marked angle is 118 degrees.

Answer: 118 degrees

4 · Reviewdoes it hold up?

Add the pentagon back up: 90 + 142 + 118 + 148 + 42 = 540, which is the 540 a pentagon must have.

Another way: The whole seven-sided figure works too: its angles add to (7 - 2) x 180 = 900 degrees. Doubling the mirrored pairs gives 2 x (142 + 118 + 148) + 84 = 900 degrees -- the same 900.

Standardsmin grade 8

4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Copying angles from the bottom half to the top half.
4.MD.C.7 Recognize angle measure as additive and solve for unknown angles — Turning exterior marks into interior angles and halving at E.
8.G.A.5 Use informal arguments to establish facts about angles of triangles and polygons — The 540 degree angle sum of a pentagon.

💡Takeaway. Fold a symmetric figure along its axis and every angle you can see on one side tells you the matching one on the other.

Variant 10 hard answer: 115 degrees

The figure on the right is a line-symmetric figure. Find the measure, in degrees, of the marked angle.

The figure is symmetric about the horizontal line through the right-end vertex $E$ (its axis of symmetry). Going around the figure, the vertices are $S$ (top), $F$ , the right-end vertex $E$ , then $D$ , $C$ , $B$ along the bottom, and the left-end vertex $A$ . The marked angles measure $90^\circ$ at vertex $E$ , $150^\circ$ at the lower vertex $D$ , and $40^\circ$ at the lower-left vertex $B$ . The angle to find is the marked angle at the top vertex $S$ .

Show solution

1 · Understandwhat's really being asked

A figure with a horizontal axis of symmetry has three marked angles (90 at E, 150 at D, 40 at B). We need the angle at the top vertex S.

Givens

An exterior mark of 90 degrees at E, which lies on the axis.
The interior angle at D is 150 degrees.
An exterior mark of 40 degrees at B.

Unknowns

The marked angle at S.

Constraints

The axis reflects the top half onto the bottom half, so matching vertices have equal angles.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Cut the figure along the axis. The top half is a pentagon, and every angle in it except the one we want can be read off using symmetry. Then the pentagon's angle sum finishes it.

3 · Execute6 carry out the plan

1Draw the axis and split off the top pentagon

#1 Draw a Diagram 4.G.A.3

Draw the axis of symmetry through O and E. It divides the figure into a top half and a bottom half that are mirror images. The top half is the five-cornered figure O-A-S-F-E.

\text{top half} = \text{pentagon } O\text{-}A\text{-}S\text{-}F\text{-}E

One pentagon instead of a seven-sided figure.

2Find the angle at E using the 90 degree mark

#7 Identify Subproblems 4.MD.C.7

The 90 degree mark at E is between side FE and the straight extension of the bottom side, so the interior angle of the whole figure at E is 180 - 90 = 90 degrees. E sits on the axis, and the axis splits that angle into two equal halves, so the pentagon's angle at E is half of it.

180^{\circ} - 90^{\circ} = 90^{\circ}, \quad 90^{\circ} \div 2 = 45^{\circ}

45 degrees at E, inside the pentagon.

3Find the angle at A using the 40 degree mark

#1 Draw a Diagram 4.G.A.3

The 40 degree mark at B is outside the figure, so the interior angle at B is 180 - 40 = 140 degrees. Because A is the mirror image of B, the corresponding angle at A is the same 140 degrees.

180^{\circ} - 40^{\circ} = 140^{\circ}

140 degrees at A.

4Find the angle at F by corresponding angles

#1 Draw a Diagram 4.G.A.3

Vertex F at the top corresponds to vertex D at the bottom, whose interior angle is given as 150 degrees. So the angle at F is also 150 degrees.

\angle SFE = \angle CDE = 150^{\circ}

150 degrees at F.

5Use the right angle on the axis at O

#1 Draw a Diagram 4.G.A.3

The segment AB joins A to its mirror image B, so it crosses the axis at a right angle. That makes the angle A-O-E inside the top pentagon a right angle.

\angle AOE = 90^{\circ}

90 degrees at O, for free.

6Subtract from the pentagon's 540 degrees

#7 Identify Subproblems 8.G.A.5

The five angles of a five-sided figure always add to 540 degrees (it splits into three triangles, 3 x 180). The top pentagon's angles are 90 at O, 140 at A, 150 at F and 45 at E, so what is left over is the angle at S.

540^{\circ} - 90^{\circ} - 140^{\circ} - 150^{\circ} - 45^{\circ} = 115^{\circ}

The marked angle is 115 degrees.

Answer: 115 degrees

4 · Reviewdoes it hold up?

Add the pentagon back up: 90 + 140 + 115 + 150 + 45 = 540, which is the 540 a pentagon must have.

Another way: The whole seven-sided figure works too: its angles add to (7 - 2) x 180 = 900 degrees. Doubling the mirrored pairs gives 2 x (140 + 115 + 150) + 90 = 900 degrees -- the same 900.

Standardsmin grade 8

4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Copying angles from the bottom half to the top half.
4.MD.C.7 Recognize angle measure as additive and solve for unknown angles — Turning exterior marks into interior angles and halving at E.
8.G.A.5 Use informal arguments to establish facts about angles of triangles and polygons — The 540 degree angle sum of a pentagon.

💡Takeaway. Fold a symmetric figure along its axis and every angle you can see on one side tells you the matching one on the other.

Variant 11 hard answer: 110 degrees

The figure on the right is a line-symmetric figure. Find the measure, in degrees, of the marked angle.

The figure is symmetric about the horizontal line through the right-end vertex $E$ (its axis of symmetry). Going around the figure, the vertices are $S$ (top), $F$ , the right-end vertex $E$ , then $D$ , $C$ , $B$ along the bottom, and the left-end vertex $A$ . The marked angles measure $110^\circ$ at vertex $E$ , $155^\circ$ at the lower vertex $D$ , and $30^\circ$ at the lower-left vertex $B$ . The angle to find is the marked angle at the top vertex $S$ .

Show solution

1 · Understandwhat's really being asked

A figure with a horizontal axis of symmetry has three marked angles (110 at E, 155 at D, 30 at B). We need the angle at the top vertex S.

Givens

An exterior mark of 110 degrees at E, which lies on the axis.
The interior angle at D is 155 degrees.
An exterior mark of 30 degrees at B.

Unknowns

The marked angle at S.

Constraints

The axis reflects the top half onto the bottom half, so matching vertices have equal angles.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Cut the figure along the axis. The top half is a pentagon, and every angle in it except the one we want can be read off using symmetry. Then the pentagon's angle sum finishes it.

3 · Execute6 carry out the plan

1Draw the axis and split off the top pentagon

#1 Draw a Diagram 4.G.A.3

Draw the axis of symmetry through O and E. It divides the figure into a top half and a bottom half that are mirror images. The top half is the five-cornered figure O-A-S-F-E.

\text{top half} = \text{pentagon } O\text{-}A\text{-}S\text{-}F\text{-}E

One pentagon instead of a seven-sided figure.

2Find the angle at E using the 110 degree mark

#7 Identify Subproblems 4.MD.C.7

The 110 degree mark at E is between side FE and the straight extension of the bottom side, so the interior angle of the whole figure at E is 180 - 110 = 70 degrees. E sits on the axis, and the axis splits that angle into two equal halves, so the pentagon's angle at E is half of it.

180^{\circ} - 110^{\circ} = 70^{\circ}, \quad 70^{\circ} \div 2 = 35^{\circ}

35 degrees at E, inside the pentagon.

3Find the angle at A using the 30 degree mark

#1 Draw a Diagram 4.G.A.3

The 30 degree mark at B is outside the figure, so the interior angle at B is 180 - 30 = 150 degrees. Because A is the mirror image of B, the corresponding angle at A is the same 150 degrees.

180^{\circ} - 30^{\circ} = 150^{\circ}

150 degrees at A.

4Find the angle at F by corresponding angles

#1 Draw a Diagram 4.G.A.3

Vertex F at the top corresponds to vertex D at the bottom, whose interior angle is given as 155 degrees. So the angle at F is also 155 degrees.

\angle SFE = \angle CDE = 155^{\circ}

155 degrees at F.

5Use the right angle on the axis at O

#1 Draw a Diagram 4.G.A.3

The segment AB joins A to its mirror image B, so it crosses the axis at a right angle. That makes the angle A-O-E inside the top pentagon a right angle.

\angle AOE = 90^{\circ}

90 degrees at O, for free.

6Subtract from the pentagon's 540 degrees

#7 Identify Subproblems 8.G.A.5

The five angles of a five-sided figure always add to 540 degrees (it splits into three triangles, 3 x 180). The top pentagon's angles are 90 at O, 150 at A, 155 at F and 35 at E, so what is left over is the angle at S.

540^{\circ} - 90^{\circ} - 150^{\circ} - 155^{\circ} - 35^{\circ} = 110^{\circ}

The marked angle is 110 degrees.

Answer: 110 degrees

4 · Reviewdoes it hold up?

Add the pentagon back up: 90 + 150 + 110 + 155 + 35 = 540, which is the 540 a pentagon must have.

Another way: The whole seven-sided figure works too: its angles add to (7 - 2) x 180 = 900 degrees. Doubling the mirrored pairs gives 2 x (150 + 110 + 155) + 70 = 900 degrees -- the same 900.

Standardsmin grade 8

4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Copying angles from the bottom half to the top half.
4.MD.C.7 Recognize angle measure as additive and solve for unknown angles — Turning exterior marks into interior angles and halving at E.
8.G.A.5 Use informal arguments to establish facts about angles of triangles and polygons — The 540 degree angle sum of a pentagon.

💡Takeaway. Fold a symmetric figure along its axis and every angle you can see on one side tells you the matching one on the other.

Variant 12 hard answer: 130 degrees

The figure on the right is a line-symmetric figure. Find the measure, in degrees, of the marked angle.

The figure is symmetric about the horizontal line through the right-end vertex $E$ (its axis of symmetry). Going around the figure, the vertices are $S$ (top), $F$ , the right-end vertex $E$ , then $D$ , $C$ , $B$ along the bottom, and the left-end vertex $A$ . The marked angles measure $120^\circ$ at vertex $E$ , $160^\circ$ at the lower vertex $D$ , and $50^\circ$ at the lower-left vertex $B$ . The angle to find is the marked angle at the top vertex $S$ .

Show solution

1 · Understandwhat's really being asked

A figure with a horizontal axis of symmetry has three marked angles (120 at E, 160 at D, 50 at B). We need the angle at the top vertex S.

Givens

An exterior mark of 120 degrees at E, which lies on the axis.
The interior angle at D is 160 degrees.
An exterior mark of 50 degrees at B.

Unknowns

The marked angle at S.

Constraints

The axis reflects the top half onto the bottom half, so matching vertices have equal angles.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Cut the figure along the axis. The top half is a pentagon, and every angle in it except the one we want can be read off using symmetry. Then the pentagon's angle sum finishes it.

3 · Execute6 carry out the plan

1Draw the axis and split off the top pentagon

#1 Draw a Diagram 4.G.A.3

Draw the axis of symmetry through O and E. It divides the figure into a top half and a bottom half that are mirror images. The top half is the five-cornered figure O-A-S-F-E.

\text{top half} = \text{pentagon } O\text{-}A\text{-}S\text{-}F\text{-}E

One pentagon instead of a seven-sided figure.

2Find the angle at E using the 120 degree mark

#7 Identify Subproblems 4.MD.C.7

The 120 degree mark at E is between side FE and the straight extension of the bottom side, so the interior angle of the whole figure at E is 180 - 120 = 60 degrees. E sits on the axis, and the axis splits that angle into two equal halves, so the pentagon's angle at E is half of it.

180^{\circ} - 120^{\circ} = 60^{\circ}, \quad 60^{\circ} \div 2 = 30^{\circ}

30 degrees at E, inside the pentagon.

3Find the angle at A using the 50 degree mark

#1 Draw a Diagram 4.G.A.3

The 50 degree mark at B is outside the figure, so the interior angle at B is 180 - 50 = 130 degrees. Because A is the mirror image of B, the corresponding angle at A is the same 130 degrees.

180^{\circ} - 50^{\circ} = 130^{\circ}

130 degrees at A.

4Find the angle at F by corresponding angles

#1 Draw a Diagram 4.G.A.3

Vertex F at the top corresponds to vertex D at the bottom, whose interior angle is given as 160 degrees. So the angle at F is also 160 degrees.

\angle SFE = \angle CDE = 160^{\circ}

160 degrees at F.

5Use the right angle on the axis at O

#1 Draw a Diagram 4.G.A.3

The segment AB joins A to its mirror image B, so it crosses the axis at a right angle. That makes the angle A-O-E inside the top pentagon a right angle.

\angle AOE = 90^{\circ}

90 degrees at O, for free.

6Subtract from the pentagon's 540 degrees

#7 Identify Subproblems 8.G.A.5

The five angles of a five-sided figure always add to 540 degrees (it splits into three triangles, 3 x 180). The top pentagon's angles are 90 at O, 130 at A, 160 at F and 30 at E, so what is left over is the angle at S.

540^{\circ} - 90^{\circ} - 130^{\circ} - 160^{\circ} - 30^{\circ} = 130^{\circ}

The marked angle is 130 degrees.

Answer: 130 degrees

4 · Reviewdoes it hold up?

Add the pentagon back up: 90 + 130 + 130 + 160 + 30 = 540, which is the 540 a pentagon must have.

Another way: The whole seven-sided figure works too: its angles add to (7 - 2) x 180 = 900 degrees. Doubling the mirrored pairs gives 2 x (130 + 130 + 160) + 60 = 900 degrees -- the same 900.

Standardsmin grade 8

4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Copying angles from the bottom half to the top half.
4.MD.C.7 Recognize angle measure as additive and solve for unknown angles — Turning exterior marks into interior angles and halving at E.
8.G.A.5 Use informal arguments to establish facts about angles of triangles and polygons — The 540 degree angle sum of a pentagon.

💡Takeaway. Fold a symmetric figure along its axis and every angle you can see on one side tells you the matching one on the other.