Practice · Draw an auxiliary parallel line to find a bent angle · Sensim Math

Variant 1 easy

Segment $\overline{AB}$ is parallel to segment $\overline{DE}$ . Find the measure of angle (circle).

Segment $\overline{AB}$ lies horizontally on the left. At point $B$ the path bends downward to point $C$ , and the interior angle at $B$ measures $120^\circ$ . At point $C$ the path bends back upward to point $D$ , and the interior angle at that vertex is the marked angle. From point $D$ a horizontal segment $\overline{DE}$ extends to the right, and the interior angle at $D$ measures $110^\circ$ . Segment $\overline{AB}$ is parallel to segment $\overline{DE}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A path goes from A to B (horizontal), bends down to C, bends back up to D, then runs horizontally to E. The top segment AB is parallel to the top segment DE. The interior angle at B is 120 deg and at D is 110 deg. I need the interior angle at C (marked with a circle).

Givens

Segment AB is parallel to segment DE.
The interior angle at B is 120 deg.
The interior angle at D is 110 deg.
AB is horizontal on the left, DE is horizontal on the right, and C is the low point between them.

Unknowns

The measure of the angle at C (the circle).

Constraints

Angles on a straight line, or co-interior angles between parallel lines, are used.
All four points form one continuous zigzag path with AB and DE parallel.

2 · Planchoose the strategy

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

Draw an auxiliary line through C that is parallel to both AB and DE. This splits the angle at C into two pieces, each a co-interior (same-side) angle with one of the parallel segments. Each piece is then 180 deg minus the given angle, and adding them gives the full angle at C.

3 · Execute4 carry out the plan

1Draw an auxiliary parallel line through C

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

Through C, draw a horizontal line parallel to AB and DE. The angle at C is split into a left part (between CB and the new line) and a right part (between the new line and CD).

\angle C = \angle(\text{left part}) + \angle(\text{right part})

Sliding a parallel line through the corner lets each side of the bend talk to a line it is parallel to, turning one hard angle into two friendly ones.

2Left part using AB

#7 Identify Subproblems 4.MD.C.7

BC crosses the parallel lines AB and the auxiliary line. The interior angle at B is 120 deg, so the co-interior angle (left part at C) on the same side is 180 deg - 120 deg.

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

Between two parallel lines, the two same-side angles add to a straight 180 deg, so the partner of 120 deg is 60 deg.

3Right part using DE

#7 Identify Subproblems 4.MD.C.7

CD crosses the auxiliary line and DE. The interior angle at D is 110 deg, so the co-interior angle (right part at C) is 180 deg - 110 deg.

180^\circ - 110^\circ = 70^\circ

Same idea on the right: the same-side partner of 110 deg is 70 deg.

4Add the two parts

#7 Identify Subproblems 4.MD.C.7

The angle at C is the sum of its left and right parts.

60^\circ + 70^\circ = 130^\circ

Angle measures simply add when two angles sit side by side around the same vertex.

Answer: 130 degrees

4 · Reviewdoes it hold up?

C is a downward-opening bend, so the answer fits the picture. The two pieces 60 deg and 70 deg are each less than the straight 180 deg they came from, and their sum 130 deg is a believable angle.

Another way: Use Identify Subproblems with a triangle: form a triangle from BC and DC and confirm the interior angle at C via the angle sum.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Drawing the auxiliary line through C parallel to AB and DE.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Finding 180-120 and 180-110, then adding 60 and 70 to get the angle at C.

💡Takeaway. Slide a parallel line through the bend and the tricky angle splits into two easy 'straight-line leftovers' you just add up!

Variant 2 easy

Segment $\overline{AB}$ is parallel to segment $\overline{DE}$ . Find the measure of angle (circle).

Segment $\overline{AB}$ lies horizontally on the left. At point $B$ the path bends downward to point $C$ , and the interior angle at $B$ measures $120^\circ$ . At point $C$ the path bends back upward to point $D$ , and the interior angle at that vertex is the marked angle. From point $D$ a horizontal segment $\overline{DE}$ extends to the right, and the interior angle at $D$ measures $115^\circ$ . Segment $\overline{AB}$ is parallel to segment $\overline{DE}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A path goes from A to B (horizontal), bends down to C, bends back up to D, then runs horizontally to E. The top segment AB is parallel to the top segment DE. The interior angle at B is 120 deg and at D is 115 deg. I need the interior angle at C (marked with a circle).

Givens

Segment AB is parallel to segment DE.
The interior angle at B is 120 deg.
The interior angle at D is 115 deg.
AB is horizontal on the left, DE is horizontal on the right, and C is the low point between them.

Unknowns

The measure of the angle at C (the circle).

Constraints

Angles on a straight line, or co-interior angles between parallel lines, are used.
All four points form one continuous zigzag path with AB and DE parallel.

2 · Planchoose the strategy

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

Draw an auxiliary line through C that is parallel to both AB and DE. This splits the angle at C into two pieces, each a co-interior (same-side) angle with one of the parallel segments. Each piece is then 180 deg minus the given angle, and adding them gives the full angle at C.

3 · Execute4 carry out the plan

1Draw an auxiliary parallel line through C

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

Through C, draw a horizontal line parallel to AB and DE. The angle at C is split into a left part (between CB and the new line) and a right part (between the new line and CD).

\angle C = \angle(\text{left part}) + \angle(\text{right part})

Sliding a parallel line through the corner lets each side of the bend talk to a line it is parallel to, turning one hard angle into two friendly ones.

2Left part using AB

#7 Identify Subproblems 4.MD.C.7

BC crosses the parallel lines AB and the auxiliary line. The interior angle at B is 120 deg, so the co-interior angle (left part at C) on the same side is 180 deg - 120 deg.

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

Between two parallel lines, the two same-side angles add to a straight 180 deg, so the partner of 120 deg is 60 deg.

3Right part using DE

#7 Identify Subproblems 4.MD.C.7

CD crosses the auxiliary line and DE. The interior angle at D is 115 deg, so the co-interior angle (right part at C) is 180 deg - 115 deg.

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

Same idea on the right: the same-side partner of 115 deg is 65 deg.

4Add the two parts

#7 Identify Subproblems 4.MD.C.7

The angle at C is the sum of its left and right parts.

60^\circ + 65^\circ = 125^\circ

Angle measures simply add when two angles sit side by side around the same vertex.

Answer: 125 degrees

4 · Reviewdoes it hold up?

C is a downward-opening bend, so the answer fits the picture. The two pieces 60 deg and 65 deg are each less than the straight 180 deg they came from, and their sum 125 deg is a believable angle.

Another way: Use Identify Subproblems with a triangle: form a triangle from BC and DC and confirm the interior angle at C via the angle sum.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Drawing the auxiliary line through C parallel to AB and DE.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Finding 180-120 and 180-115, then adding 60 and 65 to get the angle at C.

💡Takeaway. Slide a parallel line through the bend and the tricky angle splits into two easy 'straight-line leftovers' you just add up!

Variant 3 easy

Segment $\overline{AB}$ is parallel to segment $\overline{DE}$ . Find the measure of angle (circle).

Segment $\overline{AB}$ lies horizontally on the left. At point $B$ the path bends downward to point $C$ , and the interior angle at $B$ measures $115^\circ$ . At point $C$ the path bends back upward to point $D$ , and the interior angle at that vertex is the marked angle. From point $D$ a horizontal segment $\overline{DE}$ extends to the right, and the interior angle at $D$ measures $125^\circ$ . Segment $\overline{AB}$ is parallel to segment $\overline{DE}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A path goes from A to B (horizontal), bends down to C, bends back up to D, then runs horizontally to E. The top segment AB is parallel to the top segment DE. The interior angle at B is 115 deg and at D is 125 deg. I need the interior angle at C (marked with a circle).

Givens

Segment AB is parallel to segment DE.
The interior angle at B is 115 deg.
The interior angle at D is 125 deg.
AB is horizontal on the left, DE is horizontal on the right, and C is the low point between them.

Unknowns

The measure of the angle at C (the circle).

Constraints

Angles on a straight line, or co-interior angles between parallel lines, are used.
All four points form one continuous zigzag path with AB and DE parallel.

2 · Planchoose the strategy

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

Draw an auxiliary line through C that is parallel to both AB and DE. This splits the angle at C into two pieces, each a co-interior (same-side) angle with one of the parallel segments. Each piece is then 180 deg minus the given angle, and adding them gives the full angle at C.

3 · Execute4 carry out the plan

1Draw an auxiliary parallel line through C

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

Through C, draw a horizontal line parallel to AB and DE. The angle at C is split into a left part (between CB and the new line) and a right part (between the new line and CD).

\angle C = \angle(\text{left part}) + \angle(\text{right part})

Sliding a parallel line through the corner lets each side of the bend talk to a line it is parallel to, turning one hard angle into two friendly ones.

2Left part using AB

#7 Identify Subproblems 4.MD.C.7

BC crosses the parallel lines AB and the auxiliary line. The interior angle at B is 115 deg, so the co-interior angle (left part at C) on the same side is 180 deg - 115 deg.

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

Between two parallel lines, the two same-side angles add to a straight 180 deg, so the partner of 115 deg is 65 deg.

3Right part using DE

#7 Identify Subproblems 4.MD.C.7

CD crosses the auxiliary line and DE. The interior angle at D is 125 deg, so the co-interior angle (right part at C) is 180 deg - 125 deg.

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

Same idea on the right: the same-side partner of 125 deg is 55 deg.

4Add the two parts

#7 Identify Subproblems 4.MD.C.7

The angle at C is the sum of its left and right parts.

65^\circ + 55^\circ = 120^\circ

Angle measures simply add when two angles sit side by side around the same vertex.

Answer: 120 degrees

4 · Reviewdoes it hold up?

C is a downward-opening bend, so the answer fits the picture. The two pieces 65 deg and 55 deg are each less than the straight 180 deg they came from, and their sum 120 deg is a believable angle.

Another way: Use Identify Subproblems with a triangle: form a triangle from BC and DC and confirm the interior angle at C via the angle sum.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Drawing the auxiliary line through C parallel to AB and DE.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Finding 180-115 and 180-125, then adding 65 and 55 to get the angle at C.

💡Takeaway. Slide a parallel line through the bend and the tricky angle splits into two easy 'straight-line leftovers' you just add up!

Variant 4 medium

Segment $\overline{AB}$ is parallel to segment $\overline{DE}$ . Find the measure of angle (circle).

Segment $\overline{AB}$ lies horizontally on the left. At point $B$ the path bends downward to point $C$ , and the interior angle at $B$ measures $130^\circ$ . At point $C$ the path bends back upward to point $D$ , and the interior angle at that vertex is the marked angle. From point $D$ a horizontal segment $\overline{DE}$ extends to the right, and the interior angle at $D$ measures $120^\circ$ . Segment $\overline{AB}$ is parallel to segment $\overline{DE}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A path goes from A to B (horizontal), bends down to C, bends back up to D, then runs horizontally to E. The top segment AB is parallel to the top segment DE. The interior angle at B is 130 deg and at D is 120 deg. I need the interior angle at C (marked with a circle).

Givens

Segment AB is parallel to segment DE.
The interior angle at B is 130 deg.
The interior angle at D is 120 deg.
AB is horizontal on the left, DE is horizontal on the right, and C is the low point between them.

Unknowns

The measure of the angle at C (the circle).

Constraints

Angles on a straight line, or co-interior angles between parallel lines, are used.
All four points form one continuous zigzag path with AB and DE parallel.

2 · Planchoose the strategy

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

Draw an auxiliary line through C that is parallel to both AB and DE. This splits the angle at C into two pieces, each a co-interior (same-side) angle with one of the parallel segments. Each piece is then 180 deg minus the given angle, and adding them gives the full angle at C.

3 · Execute4 carry out the plan

1Draw an auxiliary parallel line through C

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

Through C, draw a horizontal line parallel to AB and DE. The angle at C is split into a left part (between CB and the new line) and a right part (between the new line and CD).

\angle C = \angle(\text{left part}) + \angle(\text{right part})

Sliding a parallel line through the corner lets each side of the bend talk to a line it is parallel to, turning one hard angle into two friendly ones.

2Left part using AB

#7 Identify Subproblems 4.MD.C.7

BC crosses the parallel lines AB and the auxiliary line. The interior angle at B is 130 deg, so the co-interior angle (left part at C) on the same side is 180 deg - 130 deg.

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

Between two parallel lines, the two same-side angles add to a straight 180 deg, so the partner of 130 deg is 50 deg.

3Right part using DE

#7 Identify Subproblems 4.MD.C.7

CD crosses the auxiliary line and DE. The interior angle at D is 120 deg, so the co-interior angle (right part at C) is 180 deg - 120 deg.

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

Same idea on the right: the same-side partner of 120 deg is 60 deg.

4Add the two parts

#7 Identify Subproblems 4.MD.C.7

The angle at C is the sum of its left and right parts.

50^\circ + 60^\circ = 110^\circ

Angle measures simply add when two angles sit side by side around the same vertex.

Answer: 110 degrees

4 · Reviewdoes it hold up?

C is a downward-opening bend, so the answer fits the picture. The two pieces 50 deg and 60 deg are each less than the straight 180 deg they came from, and their sum 110 deg is a believable angle.

Another way: Use Identify Subproblems with a triangle: form a triangle from BC and DC and confirm the interior angle at C via the angle sum.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Drawing the auxiliary line through C parallel to AB and DE.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Finding 180-130 and 180-120, then adding 50 and 60 to get the angle at C.

💡Takeaway. Slide a parallel line through the bend and the tricky angle splits into two easy 'straight-line leftovers' you just add up!

Variant 5 medium

Segment $\overline{AB}$ is parallel to segment $\overline{DE}$ . Find the measure of angle (circle).

Segment $\overline{AB}$ lies horizontally on the left. At point $B$ the path bends downward to point $C$ , and the interior angle at $B$ measures $125^\circ$ . At point $C$ the path bends back upward to point $D$ , and the interior angle at that vertex is the marked angle. From point $D$ a horizontal segment $\overline{DE}$ extends to the right, and the interior angle at $D$ measures $135^\circ$ . Segment $\overline{AB}$ is parallel to segment $\overline{DE}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A path goes from A to B (horizontal), bends down to C, bends back up to D, then runs horizontally to E. The top segment AB is parallel to the top segment DE. The interior angle at B is 125 deg and at D is 135 deg. I need the interior angle at C (marked with a circle).

Givens

Segment AB is parallel to segment DE.
The interior angle at B is 125 deg.
The interior angle at D is 135 deg.
AB is horizontal on the left, DE is horizontal on the right, and C is the low point between them.

Unknowns

The measure of the angle at C (the circle).

Constraints

Angles on a straight line, or co-interior angles between parallel lines, are used.
All four points form one continuous zigzag path with AB and DE parallel.

2 · Planchoose the strategy

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

Draw an auxiliary line through C that is parallel to both AB and DE. This splits the angle at C into two pieces, each a co-interior (same-side) angle with one of the parallel segments. Each piece is then 180 deg minus the given angle, and adding them gives the full angle at C.

3 · Execute4 carry out the plan

1Draw an auxiliary parallel line through C

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

Through C, draw a horizontal line parallel to AB and DE. The angle at C is split into a left part (between CB and the new line) and a right part (between the new line and CD).

\angle C = \angle(\text{left part}) + \angle(\text{right part})

Sliding a parallel line through the corner lets each side of the bend talk to a line it is parallel to, turning one hard angle into two friendly ones.

2Left part using AB

#7 Identify Subproblems 4.MD.C.7

BC crosses the parallel lines AB and the auxiliary line. The interior angle at B is 125 deg, so the co-interior angle (left part at C) on the same side is 180 deg - 125 deg.

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

Between two parallel lines, the two same-side angles add to a straight 180 deg, so the partner of 125 deg is 55 deg.

3Right part using DE

#7 Identify Subproblems 4.MD.C.7

CD crosses the auxiliary line and DE. The interior angle at D is 135 deg, so the co-interior angle (right part at C) is 180 deg - 135 deg.

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

Same idea on the right: the same-side partner of 135 deg is 45 deg.

4Add the two parts

#7 Identify Subproblems 4.MD.C.7

The angle at C is the sum of its left and right parts.

55^\circ + 45^\circ = 100^\circ

Angle measures simply add when two angles sit side by side around the same vertex.

Answer: 100 degrees

4 · Reviewdoes it hold up?

C is a downward-opening bend, so the answer fits the picture. The two pieces 55 deg and 45 deg are each less than the straight 180 deg they came from, and their sum 100 deg is a believable angle.

Another way: Use Identify Subproblems with a triangle: form a triangle from BC and DC and confirm the interior angle at C via the angle sum.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Drawing the auxiliary line through C parallel to AB and DE.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Finding 180-125 and 180-135, then adding 55 and 45 to get the angle at C.

💡Takeaway. Slide a parallel line through the bend and the tricky angle splits into two easy 'straight-line leftovers' you just add up!

Variant 6 medium

Segment $\overline{AB}$ is parallel to segment $\overline{DE}$ . Find the measure of angle (circle).

Segment $\overline{AB}$ lies horizontally on the left. At point $B$ the path bends downward to point $C$ , and the interior angle at $B$ measures $100^\circ$ . At point $C$ the path bends back upward to point $D$ , and the interior angle at that vertex is the marked angle. From point $D$ a horizontal segment $\overline{DE}$ extends to the right, and the interior angle at $D$ measures $140^\circ$ . Segment $\overline{AB}$ is parallel to segment $\overline{DE}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A path goes from A to B (horizontal), bends down to C, bends back up to D, then runs horizontally to E. The top segment AB is parallel to the top segment DE. The interior angle at B is 100 deg and at D is 140 deg. I need the interior angle at C (marked with a circle).

Givens

Segment AB is parallel to segment DE.
The interior angle at B is 100 deg.
The interior angle at D is 140 deg.
AB is horizontal on the left, DE is horizontal on the right, and C is the low point between them.

Unknowns

The measure of the angle at C (the circle).

Constraints

Angles on a straight line, or co-interior angles between parallel lines, are used.
All four points form one continuous zigzag path with AB and DE parallel.

2 · Planchoose the strategy

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

Draw an auxiliary line through C that is parallel to both AB and DE. This splits the angle at C into two pieces, each a co-interior (same-side) angle with one of the parallel segments. Each piece is then 180 deg minus the given angle, and adding them gives the full angle at C.

3 · Execute4 carry out the plan

1Draw an auxiliary parallel line through C

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

Through C, draw a horizontal line parallel to AB and DE. The angle at C is split into a left part (between CB and the new line) and a right part (between the new line and CD).

\angle C = \angle(\text{left part}) + \angle(\text{right part})

Sliding a parallel line through the corner lets each side of the bend talk to a line it is parallel to, turning one hard angle into two friendly ones.

2Left part using AB

#7 Identify Subproblems 4.MD.C.7

BC crosses the parallel lines AB and the auxiliary line. The interior angle at B is 100 deg, so the co-interior angle (left part at C) on the same side is 180 deg - 100 deg.

180^\circ - 100^\circ = 80^\circ

Between two parallel lines, the two same-side angles add to a straight 180 deg, so the partner of 100 deg is 80 deg.

3Right part using DE

#7 Identify Subproblems 4.MD.C.7

CD crosses the auxiliary line and DE. The interior angle at D is 140 deg, so the co-interior angle (right part at C) is 180 deg - 140 deg.

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

Same idea on the right: the same-side partner of 140 deg is 40 deg.

4Add the two parts

#7 Identify Subproblems 4.MD.C.7

The angle at C is the sum of its left and right parts.

80^\circ + 40^\circ = 120^\circ

Angle measures simply add when two angles sit side by side around the same vertex.

Answer: 120 degrees

4 · Reviewdoes it hold up?

C is a downward-opening bend, so the answer fits the picture. The two pieces 80 deg and 40 deg are each less than the straight 180 deg they came from, and their sum 120 deg is a believable angle.

Another way: Use Identify Subproblems with a triangle: form a triangle from BC and DC and confirm the interior angle at C via the angle sum.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Drawing the auxiliary line through C parallel to AB and DE.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Finding 180-100 and 180-140, then adding 80 and 40 to get the angle at C.

💡Takeaway. Slide a parallel line through the bend and the tricky angle splits into two easy 'straight-line leftovers' you just add up!

Variant 7 medium

Segment $\overline{AB}$ is parallel to segment $\overline{DE}$ . Find the measure of angle (circle).

Segment $\overline{AB}$ lies horizontally on the left. At point $B$ the path bends downward to point $C$ , and the interior angle at $B$ measures $140^\circ$ . At point $C$ the path bends back upward to point $D$ , and the interior angle at that vertex is the marked angle. From point $D$ a horizontal segment $\overline{DE}$ extends to the right, and the interior angle at $D$ measures $130^\circ$ . Segment $\overline{AB}$ is parallel to segment $\overline{DE}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A path goes from A to B (horizontal), bends down to C, bends back up to D, then runs horizontally to E. The top segment AB is parallel to the top segment DE. The interior angle at B is 140 deg and at D is 130 deg. I need the interior angle at C (marked with a circle).

Givens

Segment AB is parallel to segment DE.
The interior angle at B is 140 deg.
The interior angle at D is 130 deg.
AB is horizontal on the left, DE is horizontal on the right, and C is the low point between them.

Unknowns

The measure of the angle at C (the circle).

Constraints

Angles on a straight line, or co-interior angles between parallel lines, are used.
All four points form one continuous zigzag path with AB and DE parallel.

2 · Planchoose the strategy

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

Draw an auxiliary line through C that is parallel to both AB and DE. This splits the angle at C into two pieces, each a co-interior (same-side) angle with one of the parallel segments. Each piece is then 180 deg minus the given angle, and adding them gives the full angle at C.

3 · Execute4 carry out the plan

1Draw an auxiliary parallel line through C

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

Through C, draw a horizontal line parallel to AB and DE. The angle at C is split into a left part (between CB and the new line) and a right part (between the new line and CD).

\angle C = \angle(\text{left part}) + \angle(\text{right part})

Sliding a parallel line through the corner lets each side of the bend talk to a line it is parallel to, turning one hard angle into two friendly ones.

2Left part using AB

#7 Identify Subproblems 4.MD.C.7

BC crosses the parallel lines AB and the auxiliary line. The interior angle at B is 140 deg, so the co-interior angle (left part at C) on the same side is 180 deg - 140 deg.

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

Between two parallel lines, the two same-side angles add to a straight 180 deg, so the partner of 140 deg is 40 deg.

3Right part using DE

#7 Identify Subproblems 4.MD.C.7

CD crosses the auxiliary line and DE. The interior angle at D is 130 deg, so the co-interior angle (right part at C) is 180 deg - 130 deg.

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

Same idea on the right: the same-side partner of 130 deg is 50 deg.

4Add the two parts

#7 Identify Subproblems 4.MD.C.7

The angle at C is the sum of its left and right parts.

40^\circ + 50^\circ = 90^\circ

Angle measures simply add when two angles sit side by side around the same vertex.

Answer: 90 degrees

4 · Reviewdoes it hold up?

C is a downward-opening bend, so the answer fits the picture. The two pieces 40 deg and 50 deg are each less than the straight 180 deg they came from, and their sum 90 deg is a believable angle.

Another way: Use Identify Subproblems with a triangle: form a triangle from BC and DC and confirm the interior angle at C via the angle sum.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Drawing the auxiliary line through C parallel to AB and DE.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Finding 180-140 and 180-130, then adding 40 and 50 to get the angle at C.

💡Takeaway. Slide a parallel line through the bend and the tricky angle splits into two easy 'straight-line leftovers' you just add up!

Variant 8 hard

Segment $\overline{AB}$ is parallel to segment $\overline{DE}$ . Find the measure of angle (circle).

Segment $\overline{AB}$ lies horizontally on the left. At point $B$ the path bends downward to point $C$ , and the interior angle at $B$ measures $110^\circ$ . At point $C$ the path bends back upward to point $D$ , and the interior angle at that vertex is the marked angle. From point $D$ a horizontal segment $\overline{DE}$ extends to the right, and the interior angle at $D$ measures $145^\circ$ . Segment $\overline{AB}$ is parallel to segment $\overline{DE}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A path goes from A to B (horizontal), bends down to C, bends back up to D, then runs horizontally to E. The top segment AB is parallel to the top segment DE. The interior angle at B is 110 deg and at D is 145 deg. I need the interior angle at C (marked with a circle).

Givens

Segment AB is parallel to segment DE.
The interior angle at B is 110 deg.
The interior angle at D is 145 deg.
AB is horizontal on the left, DE is horizontal on the right, and C is the low point between them.

Unknowns

The measure of the angle at C (the circle).

Constraints

Angles on a straight line, or co-interior angles between parallel lines, are used.
All four points form one continuous zigzag path with AB and DE parallel.

2 · Planchoose the strategy

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

Draw an auxiliary line through C that is parallel to both AB and DE. This splits the angle at C into two pieces, each a co-interior (same-side) angle with one of the parallel segments. Each piece is then 180 deg minus the given angle, and adding them gives the full angle at C.

3 · Execute4 carry out the plan

1Draw an auxiliary parallel line through C

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

Through C, draw a horizontal line parallel to AB and DE. The angle at C is split into a left part (between CB and the new line) and a right part (between the new line and CD).

\angle C = \angle(\text{left part}) + \angle(\text{right part})

Sliding a parallel line through the corner lets each side of the bend talk to a line it is parallel to, turning one hard angle into two friendly ones.

2Left part using AB

#7 Identify Subproblems 4.MD.C.7

BC crosses the parallel lines AB and the auxiliary line. The interior angle at B is 110 deg, so the co-interior angle (left part at C) on the same side is 180 deg - 110 deg.

180^\circ - 110^\circ = 70^\circ

Between two parallel lines, the two same-side angles add to a straight 180 deg, so the partner of 110 deg is 70 deg.

3Right part using DE

#7 Identify Subproblems 4.MD.C.7

CD crosses the auxiliary line and DE. The interior angle at D is 145 deg, so the co-interior angle (right part at C) is 180 deg - 145 deg.

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

Same idea on the right: the same-side partner of 145 deg is 35 deg.

4Add the two parts

#7 Identify Subproblems 4.MD.C.7

The angle at C is the sum of its left and right parts.

70^\circ + 35^\circ = 105^\circ

Angle measures simply add when two angles sit side by side around the same vertex.

Answer: 105 degrees

4 · Reviewdoes it hold up?

C is a downward-opening bend, so the answer fits the picture. The two pieces 70 deg and 35 deg are each less than the straight 180 deg they came from, and their sum 105 deg is a believable angle.

Another way: Use Identify Subproblems with a triangle: form a triangle from BC and DC and confirm the interior angle at C via the angle sum.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Drawing the auxiliary line through C parallel to AB and DE.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Finding 180-110 and 180-145, then adding 70 and 35 to get the angle at C.

💡Takeaway. Slide a parallel line through the bend and the tricky angle splits into two easy 'straight-line leftovers' you just add up!

Variant 9 hard

Segment $\overline{AB}$ is parallel to segment $\overline{DE}$ . Find the measure of angle (circle).

Segment $\overline{AB}$ lies horizontally on the left. At point $B$ the path bends downward to point $C$ , and the interior angle at $B$ measures $150^\circ$ . At point $C$ the path bends back upward to point $D$ , and the interior angle at that vertex is the marked angle. From point $D$ a horizontal segment $\overline{DE}$ extends to the right, and the interior angle at $D$ measures $100^\circ$ . Segment $\overline{AB}$ is parallel to segment $\overline{DE}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A path goes from A to B (horizontal), bends down to C, bends back up to D, then runs horizontally to E. The top segment AB is parallel to the top segment DE. The interior angle at B is 150 deg and at D is 100 deg. I need the interior angle at C (marked with a circle).

Givens

Segment AB is parallel to segment DE.
The interior angle at B is 150 deg.
The interior angle at D is 100 deg.
AB is horizontal on the left, DE is horizontal on the right, and C is the low point between them.

Unknowns

The measure of the angle at C (the circle).

Constraints

Angles on a straight line, or co-interior angles between parallel lines, are used.
All four points form one continuous zigzag path with AB and DE parallel.

2 · Planchoose the strategy

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

Draw an auxiliary line through C that is parallel to both AB and DE. This splits the angle at C into two pieces, each a co-interior (same-side) angle with one of the parallel segments. Each piece is then 180 deg minus the given angle, and adding them gives the full angle at C.

3 · Execute4 carry out the plan

1Draw an auxiliary parallel line through C

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

Through C, draw a horizontal line parallel to AB and DE. The angle at C is split into a left part (between CB and the new line) and a right part (between the new line and CD).

\angle C = \angle(\text{left part}) + \angle(\text{right part})

Sliding a parallel line through the corner lets each side of the bend talk to a line it is parallel to, turning one hard angle into two friendly ones.

2Left part using AB

#7 Identify Subproblems 4.MD.C.7

BC crosses the parallel lines AB and the auxiliary line. The interior angle at B is 150 deg, so the co-interior angle (left part at C) on the same side is 180 deg - 150 deg.

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

Between two parallel lines, the two same-side angles add to a straight 180 deg, so the partner of 150 deg is 30 deg.

3Right part using DE

#7 Identify Subproblems 4.MD.C.7

CD crosses the auxiliary line and DE. The interior angle at D is 100 deg, so the co-interior angle (right part at C) is 180 deg - 100 deg.

180^\circ - 100^\circ = 80^\circ

Same idea on the right: the same-side partner of 100 deg is 80 deg.

4Add the two parts

#7 Identify Subproblems 4.MD.C.7

The angle at C is the sum of its left and right parts.

30^\circ + 80^\circ = 110^\circ

Angle measures simply add when two angles sit side by side around the same vertex.

Answer: 110 degrees

4 · Reviewdoes it hold up?

C is a downward-opening bend, so the answer fits the picture. The two pieces 30 deg and 80 deg are each less than the straight 180 deg they came from, and their sum 110 deg is a believable angle.

Another way: Use Identify Subproblems with a triangle: form a triangle from BC and DC and confirm the interior angle at C via the angle sum.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Drawing the auxiliary line through C parallel to AB and DE.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Finding 180-150 and 180-100, then adding 30 and 80 to get the angle at C.

💡Takeaway. Slide a parallel line through the bend and the tricky angle splits into two easy 'straight-line leftovers' you just add up!

Variant 10 hard

Segment $\overline{AB}$ is parallel to segment $\overline{DE}$ . Find the measure of angle (circle).

Segment $\overline{AB}$ lies horizontally on the left. At point $B$ the path bends downward to point $C$ , and the interior angle at $B$ measures $155^\circ$ . At point $C$ the path bends back upward to point $D$ , and the interior angle at that vertex is the marked angle. From point $D$ a horizontal segment $\overline{DE}$ extends to the right, and the interior angle at $D$ measures $105^\circ$ . Segment $\overline{AB}$ is parallel to segment $\overline{DE}$ .

Your answer

Show solution

1 · Understandwhat's really being asked

A path goes from A to B (horizontal), bends down to C, bends back up to D, then runs horizontally to E. The top segment AB is parallel to the top segment DE. The interior angle at B is 155 deg and at D is 105 deg. I need the interior angle at C (marked with a circle).

Givens

Segment AB is parallel to segment DE.
The interior angle at B is 155 deg.
The interior angle at D is 105 deg.
AB is horizontal on the left, DE is horizontal on the right, and C is the low point between them.

Unknowns

The measure of the angle at C (the circle).

Constraints

Angles on a straight line, or co-interior angles between parallel lines, are used.
All four points form one continuous zigzag path with AB and DE parallel.

2 · Planchoose the strategy

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

Draw an auxiliary line through C that is parallel to both AB and DE. This splits the angle at C into two pieces, each a co-interior (same-side) angle with one of the parallel segments. Each piece is then 180 deg minus the given angle, and adding them gives the full angle at C.

3 · Execute4 carry out the plan

1Draw an auxiliary parallel line through C

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

Through C, draw a horizontal line parallel to AB and DE. The angle at C is split into a left part (between CB and the new line) and a right part (between the new line and CD).

\angle C = \angle(\text{left part}) + \angle(\text{right part})

Sliding a parallel line through the corner lets each side of the bend talk to a line it is parallel to, turning one hard angle into two friendly ones.

2Left part using AB

#7 Identify Subproblems 4.MD.C.7

BC crosses the parallel lines AB and the auxiliary line. The interior angle at B is 155 deg, so the co-interior angle (left part at C) on the same side is 180 deg - 155 deg.

180^\circ - 155^\circ = 25^\circ

Between two parallel lines, the two same-side angles add to a straight 180 deg, so the partner of 155 deg is 25 deg.

3Right part using DE

#7 Identify Subproblems 4.MD.C.7

CD crosses the auxiliary line and DE. The interior angle at D is 105 deg, so the co-interior angle (right part at C) is 180 deg - 105 deg.

180^\circ - 105^\circ = 75^\circ

Same idea on the right: the same-side partner of 105 deg is 75 deg.

4Add the two parts

#7 Identify Subproblems 4.MD.C.7

The angle at C is the sum of its left and right parts.

25^\circ + 75^\circ = 100^\circ

Angle measures simply add when two angles sit side by side around the same vertex.

Answer: 100 degrees

4 · Reviewdoes it hold up?

C is a downward-opening bend, so the answer fits the picture. The two pieces 25 deg and 75 deg are each less than the straight 180 deg they came from, and their sum 100 deg is a believable angle.

Another way: Use Identify Subproblems with a triangle: form a triangle from BC and DC and confirm the interior angle at C via the angle sum.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Drawing the auxiliary line through C parallel to AB and DE.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Finding 180-155 and 180-105, then adding 25 and 75 to get the angle at C.

💡Takeaway. Slide a parallel line through the bend and the tricky angle splits into two easy 'straight-line leftovers' you just add up!

Draw an auxiliary parallel line to find a bent angle · 10 practice problems

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