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← 4-2 · Rotation preserves side lengths and angle measures · Transformations Preserve Measures

Rotation preserves side lengths and angle measures · 10 practice problems

4.MD.C.7

Generated variants — 10

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

Variant 1 answer: 15 degrees

As shown, equilateral triangle ABCABC is rotated 75°75\degree clockwise about point AA to make equilateral triangle ADEADE. Find the measure of angle (1) (marked at point AA between side ABAB and side AEAE).

① A B C D E
Show solution

Understand

An equilateral triangle ABC is rotated 75 degrees clockwise about point A to make equilateral triangle ADE (B maps to D, C maps to E). Angle 1 is the angle at A between side AB and side AE. I must find its measure.

Givens
  • Triangle ABC is equilateral, so each of its angles is 60 degrees
  • It is rotated 75 degrees clockwise about A, so the rotation angle is 75 degrees
  • The rotation sends B to D and C to E, making equilateral triangle ADE
  • Angle 1 is the angle at A between AB and AE
Unknowns
  • The measure of angle 1 (angle BAE)
Constraints
  • A rotation keeps all side lengths and angle sizes unchanged
  • Each angle of an equilateral triangle is 60 degrees
  • The rotation turned AB to AD through exactly 75 degrees

Plan

#17 Visualize Spatial Relationships · also uses: #1 Draw a Diagram#7 Identify Subproblems

Rotation is a spatial action, so I picture how AB swings to AD by the turn and where AE lands. Then I split the turn into the part covered by the equilateral triangle's angle at A and the leftover, which is angle 1.

Execute

#17 Visualize Spatial Relationships 4.MD.C.5
Rotating 75 degrees clockwise about A sends B to D, so ray AB turns to ray AD through exactly 75 degrees: angle BAD = 75 degrees.
∠BAD=75∘\angle BAD = 75^\circ
The amount you turn the shape is exactly the angle between a point and its rotated image.
#17 Visualize Spatial Relationships 4.G.A.2
Rotation does not change angle sizes, so triangle ADE is still equilateral and its angle at A, angle DAE, is 60 degrees.
∠DAE=60∘\angle DAE = 60^\circ
Turning a shape does not stretch or bend it, so every angle stays the same size.
#7 Identify Subproblems 4.MD.C.7
Ray AE lies inside the 75-degree turn between AB and AD, with angle DAE = 60 degrees taking up part of it. So angle 1 = angle BAD - angle DAE = 75 - 60 = 15 degrees.
∠1=∠BAD−∠DAE=75∘−60∘=15∘\angle 1 = \angle BAD - \angle DAE = 75^\circ - 60^\circ = 15^\circ
Angle measure is additive, so the leftover piece of the turn after the triangle's angle is the answer.
Answer: 15 degrees

Review

Angle 1 = 15 degrees is the small leftover sliver between AE and AB, smaller than the 75-degree rotation. The arithmetic 75 - 60 = 15 is exact, so it checks out.

Draw the two triangles to scale on paper (tool 1), actually rotate a cutout of ABC about A, and measure angle 1 with a protractor to confirm.

Standards · min grade 4

  • 4.MD.C.5 Recognize angles as geometric shapes formed when two rays share an endpoint — Treating the turn as the angle between ray AB and its rotated image ray AD.
  • 4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the equilateral triangle's angle at A, preserved under rotation.
  • 4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting the triangle's angle from the rotation to find angle 1.
💡 This only needs Grade 4 angle-subtracting plus knowing a turn keeps every angle the same size!
Variant 2 answer: 20 degrees

As shown, equilateral triangle ABCABC is rotated 80°80\degree clockwise about point AA to make equilateral triangle ADEADE. Find the measure of angle (1) (marked at point AA between side ABAB and side AEAE).

① A B C D E
Show solution

Understand

An equilateral triangle ABC is rotated 80 degrees clockwise about point A to make equilateral triangle ADE (B maps to D, C maps to E). Angle 1 is the angle at A between side AB and side AE. I must find its measure.

Givens
  • Triangle ABC is equilateral, so each of its angles is 60 degrees
  • It is rotated 80 degrees clockwise about A, so the rotation angle is 80 degrees
  • The rotation sends B to D and C to E, making equilateral triangle ADE
  • Angle 1 is the angle at A between AB and AE
Unknowns
  • The measure of angle 1 (angle BAE)
Constraints
  • A rotation keeps all side lengths and angle sizes unchanged
  • Each angle of an equilateral triangle is 60 degrees
  • The rotation turned AB to AD through exactly 80 degrees

Plan

#17 Visualize Spatial Relationships · also uses: #1 Draw a Diagram#7 Identify Subproblems

Rotation is a spatial action, so I picture how AB swings to AD by the turn and where AE lands. Then I split the turn into the part covered by the equilateral triangle's angle at A and the leftover, which is angle 1.

Execute

#17 Visualize Spatial Relationships 4.MD.C.5
Rotating 80 degrees clockwise about A sends B to D, so ray AB turns to ray AD through exactly 80 degrees: angle BAD = 80 degrees.
∠BAD=80∘\angle BAD = 80^\circ
The amount you turn the shape is exactly the angle between a point and its rotated image.
#17 Visualize Spatial Relationships 4.G.A.2
Rotation does not change angle sizes, so triangle ADE is still equilateral and its angle at A, angle DAE, is 60 degrees.
∠DAE=60∘\angle DAE = 60^\circ
Turning a shape does not stretch or bend it, so every angle stays the same size.
#7 Identify Subproblems 4.MD.C.7
Ray AE lies inside the 80-degree turn between AB and AD, with angle DAE = 60 degrees taking up part of it. So angle 1 = angle BAD - angle DAE = 80 - 60 = 20 degrees.
∠1=∠BAD−∠DAE=80∘−60∘=20∘\angle 1 = \angle BAD - \angle DAE = 80^\circ - 60^\circ = 20^\circ
Angle measure is additive, so the leftover piece of the turn after the triangle's angle is the answer.
Answer: 20 degrees

Review

Angle 1 = 20 degrees is the small leftover sliver between AE and AB, smaller than the 80-degree rotation. The arithmetic 80 - 60 = 20 is exact, so it checks out.

Draw the two triangles to scale on paper (tool 1), actually rotate a cutout of ABC about A, and measure angle 1 with a protractor to confirm.

Standards · min grade 4

  • 4.MD.C.5 Recognize angles as geometric shapes formed when two rays share an endpoint — Treating the turn as the angle between ray AB and its rotated image ray AD.
  • 4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the equilateral triangle's angle at A, preserved under rotation.
  • 4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting the triangle's angle from the rotation to find angle 1.
💡 This only needs Grade 4 angle-subtracting plus knowing a turn keeps every angle the same size!
Variant 3 answer: 60 degrees

As shown, equilateral triangle ABCABC is rotated 120°120\degree clockwise about point AA to make equilateral triangle ADEADE. Find the measure of angle (1) (marked at point AA between side ABAB and side AEAE).

① A B C D E
Show solution

Understand

An equilateral triangle ABC is rotated 120 degrees clockwise about point A to make equilateral triangle ADE (B maps to D, C maps to E). Angle 1 is the angle at A between side AB and side AE. I must find its measure.

Givens
  • Triangle ABC is equilateral, so each of its angles is 60 degrees
  • It is rotated 120 degrees clockwise about A, so the rotation angle is 120 degrees
  • The rotation sends B to D and C to E, making equilateral triangle ADE
  • Angle 1 is the angle at A between AB and AE
Unknowns
  • The measure of angle 1 (angle BAE)
Constraints
  • A rotation keeps all side lengths and angle sizes unchanged
  • Each angle of an equilateral triangle is 60 degrees
  • The rotation turned AB to AD through exactly 120 degrees

Plan

#17 Visualize Spatial Relationships · also uses: #1 Draw a Diagram#7 Identify Subproblems

Rotation is a spatial action, so I picture how AB swings to AD by the turn and where AE lands. Then I split the turn into the part covered by the equilateral triangle's angle at A and the leftover, which is angle 1.

Execute

#17 Visualize Spatial Relationships 4.MD.C.5
Rotating 120 degrees clockwise about A sends B to D, so ray AB turns to ray AD through exactly 120 degrees: angle BAD = 120 degrees.
∠BAD=120∘\angle BAD = 120^\circ
The amount you turn the shape is exactly the angle between a point and its rotated image.
#17 Visualize Spatial Relationships 4.G.A.2
Rotation does not change angle sizes, so triangle ADE is still equilateral and its angle at A, angle DAE, is 60 degrees.
∠DAE=60∘\angle DAE = 60^\circ
Turning a shape does not stretch or bend it, so every angle stays the same size.
#7 Identify Subproblems 4.MD.C.7
Ray AE lies inside the 120-degree turn between AB and AD, with angle DAE = 60 degrees taking up part of it. So angle 1 = angle BAD - angle DAE = 120 - 60 = 60 degrees.
∠1=∠BAD−∠DAE=120∘−60∘=60∘\angle 1 = \angle BAD - \angle DAE = 120^\circ - 60^\circ = 60^\circ
Angle measure is additive, so the leftover piece of the turn after the triangle's angle is the answer.
Answer: 60 degrees

Review

Angle 1 = 60 degrees is the small leftover sliver between AE and AB, smaller than the 120-degree rotation. The arithmetic 120 - 60 = 60 is exact, so it checks out.

Draw the two triangles to scale on paper (tool 1), actually rotate a cutout of ABC about A, and measure angle 1 with a protractor to confirm.

Standards · min grade 4

  • 4.MD.C.5 Recognize angles as geometric shapes formed when two rays share an endpoint — Treating the turn as the angle between ray AB and its rotated image ray AD.
  • 4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the equilateral triangle's angle at A, preserved under rotation.
  • 4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting the triangle's angle from the rotation to find angle 1.
💡 This only needs Grade 4 angle-subtracting plus knowing a turn keeps every angle the same size!
Variant 4 answer: 40 degrees

As shown, equilateral triangle ABCABC is rotated 100°100\degree clockwise about point AA to make equilateral triangle ADEADE. Find the measure of angle (1) (marked at point AA between side ABAB and side AEAE).

① A B C D E
Show solution

Understand

An equilateral triangle ABC is rotated 100 degrees clockwise about point A to make equilateral triangle ADE (B maps to D, C maps to E). Angle 1 is the angle at A between side AB and side AE. I must find its measure.

Givens
  • Triangle ABC is equilateral, so each of its angles is 60 degrees
  • It is rotated 100 degrees clockwise about A, so the rotation angle is 100 degrees
  • The rotation sends B to D and C to E, making equilateral triangle ADE
  • Angle 1 is the angle at A between AB and AE
Unknowns
  • The measure of angle 1 (angle BAE)
Constraints
  • A rotation keeps all side lengths and angle sizes unchanged
  • Each angle of an equilateral triangle is 60 degrees
  • The rotation turned AB to AD through exactly 100 degrees

Plan

#17 Visualize Spatial Relationships · also uses: #1 Draw a Diagram#7 Identify Subproblems

Rotation is a spatial action, so I picture how AB swings to AD by the turn and where AE lands. Then I split the turn into the part covered by the equilateral triangle's angle at A and the leftover, which is angle 1.

Execute

#17 Visualize Spatial Relationships 4.MD.C.5
Rotating 100 degrees clockwise about A sends B to D, so ray AB turns to ray AD through exactly 100 degrees: angle BAD = 100 degrees.
∠BAD=100∘\angle BAD = 100^\circ
The amount you turn the shape is exactly the angle between a point and its rotated image.
#17 Visualize Spatial Relationships 4.G.A.2
Rotation does not change angle sizes, so triangle ADE is still equilateral and its angle at A, angle DAE, is 60 degrees.
∠DAE=60∘\angle DAE = 60^\circ
Turning a shape does not stretch or bend it, so every angle stays the same size.
#7 Identify Subproblems 4.MD.C.7
Ray AE lies inside the 100-degree turn between AB and AD, with angle DAE = 60 degrees taking up part of it. So angle 1 = angle BAD - angle DAE = 100 - 60 = 40 degrees.
∠1=∠BAD−∠DAE=100∘−60∘=40∘\angle 1 = \angle BAD - \angle DAE = 100^\circ - 60^\circ = 40^\circ
Angle measure is additive, so the leftover piece of the turn after the triangle's angle is the answer.
Answer: 40 degrees

Review

Angle 1 = 40 degrees is the small leftover sliver between AE and AB, smaller than the 100-degree rotation. The arithmetic 100 - 60 = 40 is exact, so it checks out.

Draw the two triangles to scale on paper (tool 1), actually rotate a cutout of ABC about A, and measure angle 1 with a protractor to confirm.

Standards · min grade 4

  • 4.MD.C.5 Recognize angles as geometric shapes formed when two rays share an endpoint — Treating the turn as the angle between ray AB and its rotated image ray AD.
  • 4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the equilateral triangle's angle at A, preserved under rotation.
  • 4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting the triangle's angle from the rotation to find angle 1.
💡 This only needs Grade 4 angle-subtracting plus knowing a turn keeps every angle the same size!
Variant 5 answer: 100 degrees

As shown, equilateral triangle ABCABC is rotated 160°160\degree clockwise about point AA to make equilateral triangle ADEADE. Find the measure of angle (1) (marked at point AA between side ABAB and side AEAE).

① A B C D E
Show solution

Understand

An equilateral triangle ABC is rotated 160 degrees clockwise about point A to make equilateral triangle ADE (B maps to D, C maps to E). Angle 1 is the angle at A between side AB and side AE. I must find its measure.

Givens
  • Triangle ABC is equilateral, so each of its angles is 60 degrees
  • It is rotated 160 degrees clockwise about A, so the rotation angle is 160 degrees
  • The rotation sends B to D and C to E, making equilateral triangle ADE
  • Angle 1 is the angle at A between AB and AE
Unknowns
  • The measure of angle 1 (angle BAE)
Constraints
  • A rotation keeps all side lengths and angle sizes unchanged
  • Each angle of an equilateral triangle is 60 degrees
  • The rotation turned AB to AD through exactly 160 degrees

Plan

#17 Visualize Spatial Relationships · also uses: #1 Draw a Diagram#7 Identify Subproblems

Rotation is a spatial action, so I picture how AB swings to AD by the turn and where AE lands. Then I split the turn into the part covered by the equilateral triangle's angle at A and the leftover, which is angle 1.

Execute

#17 Visualize Spatial Relationships 4.MD.C.5
Rotating 160 degrees clockwise about A sends B to D, so ray AB turns to ray AD through exactly 160 degrees: angle BAD = 160 degrees.
∠BAD=160∘\angle BAD = 160^\circ
The amount you turn the shape is exactly the angle between a point and its rotated image.
#17 Visualize Spatial Relationships 4.G.A.2
Rotation does not change angle sizes, so triangle ADE is still equilateral and its angle at A, angle DAE, is 60 degrees.
∠DAE=60∘\angle DAE = 60^\circ
Turning a shape does not stretch or bend it, so every angle stays the same size.
#7 Identify Subproblems 4.MD.C.7
Ray AE lies inside the 160-degree turn between AB and AD, with angle DAE = 60 degrees taking up part of it. So angle 1 = angle BAD - angle DAE = 160 - 60 = 100 degrees.
∠1=∠BAD−∠DAE=160∘−60∘=100∘\angle 1 = \angle BAD - \angle DAE = 160^\circ - 60^\circ = 100^\circ
Angle measure is additive, so the leftover piece of the turn after the triangle's angle is the answer.
Answer: 100 degrees

Review

Angle 1 = 100 degrees is the small leftover sliver between AE and AB, smaller than the 160-degree rotation. The arithmetic 160 - 60 = 100 is exact, so it checks out.

Draw the two triangles to scale on paper (tool 1), actually rotate a cutout of ABC about A, and measure angle 1 with a protractor to confirm.

Standards · min grade 4

  • 4.MD.C.5 Recognize angles as geometric shapes formed when two rays share an endpoint — Treating the turn as the angle between ray AB and its rotated image ray AD.
  • 4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the equilateral triangle's angle at A, preserved under rotation.
  • 4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting the triangle's angle from the rotation to find angle 1.
💡 This only needs Grade 4 angle-subtracting plus knowing a turn keeps every angle the same size!
Variant 6 answer: 50 degrees

As shown, equilateral triangle ABCABC is rotated 110°110\degree clockwise about point AA to make equilateral triangle ADEADE. Find the measure of angle (1) (marked at point AA between side ABAB and side AEAE).

① A B C D E
Show solution

Understand

An equilateral triangle ABC is rotated 110 degrees clockwise about point A to make equilateral triangle ADE (B maps to D, C maps to E). Angle 1 is the angle at A between side AB and side AE. I must find its measure.

Givens
  • Triangle ABC is equilateral, so each of its angles is 60 degrees
  • It is rotated 110 degrees clockwise about A, so the rotation angle is 110 degrees
  • The rotation sends B to D and C to E, making equilateral triangle ADE
  • Angle 1 is the angle at A between AB and AE
Unknowns
  • The measure of angle 1 (angle BAE)
Constraints
  • A rotation keeps all side lengths and angle sizes unchanged
  • Each angle of an equilateral triangle is 60 degrees
  • The rotation turned AB to AD through exactly 110 degrees

Plan

#17 Visualize Spatial Relationships · also uses: #1 Draw a Diagram#7 Identify Subproblems

Rotation is a spatial action, so I picture how AB swings to AD by the turn and where AE lands. Then I split the turn into the part covered by the equilateral triangle's angle at A and the leftover, which is angle 1.

Execute

#17 Visualize Spatial Relationships 4.MD.C.5
Rotating 110 degrees clockwise about A sends B to D, so ray AB turns to ray AD through exactly 110 degrees: angle BAD = 110 degrees.
∠BAD=110∘\angle BAD = 110^\circ
The amount you turn the shape is exactly the angle between a point and its rotated image.
#17 Visualize Spatial Relationships 4.G.A.2
Rotation does not change angle sizes, so triangle ADE is still equilateral and its angle at A, angle DAE, is 60 degrees.
∠DAE=60∘\angle DAE = 60^\circ
Turning a shape does not stretch or bend it, so every angle stays the same size.
#7 Identify Subproblems 4.MD.C.7
Ray AE lies inside the 110-degree turn between AB and AD, with angle DAE = 60 degrees taking up part of it. So angle 1 = angle BAD - angle DAE = 110 - 60 = 50 degrees.
∠1=∠BAD−∠DAE=110∘−60∘=50∘\angle 1 = \angle BAD - \angle DAE = 110^\circ - 60^\circ = 50^\circ
Angle measure is additive, so the leftover piece of the turn after the triangle's angle is the answer.
Answer: 50 degrees

Review

Angle 1 = 50 degrees is the small leftover sliver between AE and AB, smaller than the 110-degree rotation. The arithmetic 110 - 60 = 50 is exact, so it checks out.

Draw the two triangles to scale on paper (tool 1), actually rotate a cutout of ABC about A, and measure angle 1 with a protractor to confirm.

Standards · min grade 4

  • 4.MD.C.5 Recognize angles as geometric shapes formed when two rays share an endpoint — Treating the turn as the angle between ray AB and its rotated image ray AD.
  • 4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the equilateral triangle's angle at A, preserved under rotation.
  • 4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting the triangle's angle from the rotation to find angle 1.
💡 This only needs Grade 4 angle-subtracting plus knowing a turn keeps every angle the same size!
Variant 7 answer: 35 degrees

As shown, equilateral triangle ABCABC is rotated 95°95\degree clockwise about point AA to make equilateral triangle ADEADE. Find the measure of angle (1) (marked at point AA between side ABAB and side AEAE).

① A B C D E
Show solution

Understand

An equilateral triangle ABC is rotated 95 degrees clockwise about point A to make equilateral triangle ADE (B maps to D, C maps to E). Angle 1 is the angle at A between side AB and side AE. I must find its measure.

Givens
  • Triangle ABC is equilateral, so each of its angles is 60 degrees
  • It is rotated 95 degrees clockwise about A, so the rotation angle is 95 degrees
  • The rotation sends B to D and C to E, making equilateral triangle ADE
  • Angle 1 is the angle at A between AB and AE
Unknowns
  • The measure of angle 1 (angle BAE)
Constraints
  • A rotation keeps all side lengths and angle sizes unchanged
  • Each angle of an equilateral triangle is 60 degrees
  • The rotation turned AB to AD through exactly 95 degrees

Plan

#17 Visualize Spatial Relationships · also uses: #1 Draw a Diagram#7 Identify Subproblems

Rotation is a spatial action, so I picture how AB swings to AD by the turn and where AE lands. Then I split the turn into the part covered by the equilateral triangle's angle at A and the leftover, which is angle 1.

Execute

#17 Visualize Spatial Relationships 4.MD.C.5
Rotating 95 degrees clockwise about A sends B to D, so ray AB turns to ray AD through exactly 95 degrees: angle BAD = 95 degrees.
∠BAD=95∘\angle BAD = 95^\circ
The amount you turn the shape is exactly the angle between a point and its rotated image.
#17 Visualize Spatial Relationships 4.G.A.2
Rotation does not change angle sizes, so triangle ADE is still equilateral and its angle at A, angle DAE, is 60 degrees.
∠DAE=60∘\angle DAE = 60^\circ
Turning a shape does not stretch or bend it, so every angle stays the same size.
#7 Identify Subproblems 4.MD.C.7
Ray AE lies inside the 95-degree turn between AB and AD, with angle DAE = 60 degrees taking up part of it. So angle 1 = angle BAD - angle DAE = 95 - 60 = 35 degrees.
∠1=∠BAD−∠DAE=95∘−60∘=35∘\angle 1 = \angle BAD - \angle DAE = 95^\circ - 60^\circ = 35^\circ
Angle measure is additive, so the leftover piece of the turn after the triangle's angle is the answer.
Answer: 35 degrees

Review

Angle 1 = 35 degrees is the small leftover sliver between AE and AB, smaller than the 95-degree rotation. The arithmetic 95 - 60 = 35 is exact, so it checks out.

Draw the two triangles to scale on paper (tool 1), actually rotate a cutout of ABC about A, and measure angle 1 with a protractor to confirm.

Standards · min grade 4

  • 4.MD.C.5 Recognize angles as geometric shapes formed when two rays share an endpoint — Treating the turn as the angle between ray AB and its rotated image ray AD.
  • 4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the equilateral triangle's angle at A, preserved under rotation.
  • 4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting the triangle's angle from the rotation to find angle 1.
💡 This only needs Grade 4 angle-subtracting plus knowing a turn keeps every angle the same size!
Variant 8 answer: 90 degrees

As shown, equilateral triangle ABCABC is rotated 150°150\degree clockwise about point AA to make equilateral triangle ADEADE. Find the measure of angle (1) (marked at point AA between side ABAB and side AEAE).

① A B C D E
Show solution

Understand

An equilateral triangle ABC is rotated 150 degrees clockwise about point A to make equilateral triangle ADE (B maps to D, C maps to E). Angle 1 is the angle at A between side AB and side AE. I must find its measure.

Givens
  • Triangle ABC is equilateral, so each of its angles is 60 degrees
  • It is rotated 150 degrees clockwise about A, so the rotation angle is 150 degrees
  • The rotation sends B to D and C to E, making equilateral triangle ADE
  • Angle 1 is the angle at A between AB and AE
Unknowns
  • The measure of angle 1 (angle BAE)
Constraints
  • A rotation keeps all side lengths and angle sizes unchanged
  • Each angle of an equilateral triangle is 60 degrees
  • The rotation turned AB to AD through exactly 150 degrees

Plan

#17 Visualize Spatial Relationships · also uses: #1 Draw a Diagram#7 Identify Subproblems

Rotation is a spatial action, so I picture how AB swings to AD by the turn and where AE lands. Then I split the turn into the part covered by the equilateral triangle's angle at A and the leftover, which is angle 1.

Execute

#17 Visualize Spatial Relationships 4.MD.C.5
Rotating 150 degrees clockwise about A sends B to D, so ray AB turns to ray AD through exactly 150 degrees: angle BAD = 150 degrees.
∠BAD=150∘\angle BAD = 150^\circ
The amount you turn the shape is exactly the angle between a point and its rotated image.
#17 Visualize Spatial Relationships 4.G.A.2
Rotation does not change angle sizes, so triangle ADE is still equilateral and its angle at A, angle DAE, is 60 degrees.
∠DAE=60∘\angle DAE = 60^\circ
Turning a shape does not stretch or bend it, so every angle stays the same size.
#7 Identify Subproblems 4.MD.C.7
Ray AE lies inside the 150-degree turn between AB and AD, with angle DAE = 60 degrees taking up part of it. So angle 1 = angle BAD - angle DAE = 150 - 60 = 90 degrees.
∠1=∠BAD−∠DAE=150∘−60∘=90∘\angle 1 = \angle BAD - \angle DAE = 150^\circ - 60^\circ = 90^\circ
Angle measure is additive, so the leftover piece of the turn after the triangle's angle is the answer.
Answer: 90 degrees

Review

Angle 1 = 90 degrees is the small leftover sliver between AE and AB, smaller than the 150-degree rotation. The arithmetic 150 - 60 = 90 is exact, so it checks out.

Draw the two triangles to scale on paper (tool 1), actually rotate a cutout of ABC about A, and measure angle 1 with a protractor to confirm.

Standards · min grade 4

  • 4.MD.C.5 Recognize angles as geometric shapes formed when two rays share an endpoint — Treating the turn as the angle between ray AB and its rotated image ray AD.
  • 4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the equilateral triangle's angle at A, preserved under rotation.
  • 4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting the triangle's angle from the rotation to find angle 1.
💡 This only needs Grade 4 angle-subtracting plus knowing a turn keeps every angle the same size!
Variant 9 answer: 70 degrees

As shown, equilateral triangle ABCABC is rotated 130°130\degree clockwise about point AA to make equilateral triangle ADEADE. Find the measure of angle (1) (marked at point AA between side ABAB and side AEAE).

① A B C D E
Show solution

Understand

An equilateral triangle ABC is rotated 130 degrees clockwise about point A to make equilateral triangle ADE (B maps to D, C maps to E). Angle 1 is the angle at A between side AB and side AE. I must find its measure.

Givens
  • Triangle ABC is equilateral, so each of its angles is 60 degrees
  • It is rotated 130 degrees clockwise about A, so the rotation angle is 130 degrees
  • The rotation sends B to D and C to E, making equilateral triangle ADE
  • Angle 1 is the angle at A between AB and AE
Unknowns
  • The measure of angle 1 (angle BAE)
Constraints
  • A rotation keeps all side lengths and angle sizes unchanged
  • Each angle of an equilateral triangle is 60 degrees
  • The rotation turned AB to AD through exactly 130 degrees

Plan

#17 Visualize Spatial Relationships · also uses: #1 Draw a Diagram#7 Identify Subproblems

Rotation is a spatial action, so I picture how AB swings to AD by the turn and where AE lands. Then I split the turn into the part covered by the equilateral triangle's angle at A and the leftover, which is angle 1.

Execute

#17 Visualize Spatial Relationships 4.MD.C.5
Rotating 130 degrees clockwise about A sends B to D, so ray AB turns to ray AD through exactly 130 degrees: angle BAD = 130 degrees.
∠BAD=130∘\angle BAD = 130^\circ
The amount you turn the shape is exactly the angle between a point and its rotated image.
#17 Visualize Spatial Relationships 4.G.A.2
Rotation does not change angle sizes, so triangle ADE is still equilateral and its angle at A, angle DAE, is 60 degrees.
∠DAE=60∘\angle DAE = 60^\circ
Turning a shape does not stretch or bend it, so every angle stays the same size.
#7 Identify Subproblems 4.MD.C.7
Ray AE lies inside the 130-degree turn between AB and AD, with angle DAE = 60 degrees taking up part of it. So angle 1 = angle BAD - angle DAE = 130 - 60 = 70 degrees.
∠1=∠BAD−∠DAE=130∘−60∘=70∘\angle 1 = \angle BAD - \angle DAE = 130^\circ - 60^\circ = 70^\circ
Angle measure is additive, so the leftover piece of the turn after the triangle's angle is the answer.
Answer: 70 degrees

Review

Angle 1 = 70 degrees is the small leftover sliver between AE and AB, smaller than the 130-degree rotation. The arithmetic 130 - 60 = 70 is exact, so it checks out.

Draw the two triangles to scale on paper (tool 1), actually rotate a cutout of ABC about A, and measure angle 1 with a protractor to confirm.

Standards · min grade 4

  • 4.MD.C.5 Recognize angles as geometric shapes formed when two rays share an endpoint — Treating the turn as the angle between ray AB and its rotated image ray AD.
  • 4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the equilateral triangle's angle at A, preserved under rotation.
  • 4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting the triangle's angle from the rotation to find angle 1.
💡 This only needs Grade 4 angle-subtracting plus knowing a turn keeps every angle the same size!
Variant 10 answer: 30 degrees

As shown, equilateral triangle ABCABC is rotated 90°90\degree clockwise about point AA to make equilateral triangle ADEADE. Find the measure of angle (1) (marked at point AA between side ABAB and side AEAE).

① A B C D E
Show solution

Understand

An equilateral triangle ABC is rotated 90 degrees clockwise about point A to make equilateral triangle ADE (B maps to D, C maps to E). Angle 1 is the angle at A between side AB and side AE. I must find its measure.

Givens
  • Triangle ABC is equilateral, so each of its angles is 60 degrees
  • It is rotated 90 degrees clockwise about A, so the rotation angle is 90 degrees
  • The rotation sends B to D and C to E, making equilateral triangle ADE
  • Angle 1 is the angle at A between AB and AE
Unknowns
  • The measure of angle 1 (angle BAE)
Constraints
  • A rotation keeps all side lengths and angle sizes unchanged
  • Each angle of an equilateral triangle is 60 degrees
  • The rotation turned AB to AD through exactly 90 degrees

Plan

#17 Visualize Spatial Relationships · also uses: #1 Draw a Diagram#7 Identify Subproblems

Rotation is a spatial action, so I picture how AB swings to AD by the turn and where AE lands. Then I split the turn into the part covered by the equilateral triangle's angle at A and the leftover, which is angle 1.

Execute

#17 Visualize Spatial Relationships 4.MD.C.5
Rotating 90 degrees clockwise about A sends B to D, so ray AB turns to ray AD through exactly 90 degrees: angle BAD = 90 degrees.
∠BAD=90∘\angle BAD = 90^\circ
The amount you turn the shape is exactly the angle between a point and its rotated image.
#17 Visualize Spatial Relationships 4.G.A.2
Rotation does not change angle sizes, so triangle ADE is still equilateral and its angle at A, angle DAE, is 60 degrees.
∠DAE=60∘\angle DAE = 60^\circ
Turning a shape does not stretch or bend it, so every angle stays the same size.
#7 Identify Subproblems 4.MD.C.7
Ray AE lies inside the 90-degree turn between AB and AD, with angle DAE = 60 degrees taking up part of it. So angle 1 = angle BAD - angle DAE = 90 - 60 = 30 degrees.
∠1=∠BAD−∠DAE=90∘−60∘=30∘\angle 1 = \angle BAD - \angle DAE = 90^\circ - 60^\circ = 30^\circ
Angle measure is additive, so the leftover piece of the turn after the triangle's angle is the answer.
Answer: 30 degrees

Review

Angle 1 = 30 degrees is the small leftover sliver between AE and AB, smaller than the 90-degree rotation. The arithmetic 90 - 60 = 30 is exact, so it checks out.

Draw the two triangles to scale on paper (tool 1), actually rotate a cutout of ABC about A, and measure angle 1 with a protractor to confirm.

Standards · min grade 4

  • 4.MD.C.5 Recognize angles as geometric shapes formed when two rays share an endpoint — Treating the turn as the angle between ray AB and its rotated image ray AD.
  • 4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the equilateral triangle's angle at A, preserved under rotation.
  • 4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting the triangle's angle from the rotation to find angle 1.
💡 This only needs Grade 4 angle-subtracting plus knowing a turn keeps every angle the same size!