Practice · Rotation preserves side lengths and angle measures · Sensim Math

Variant 1 easy

As shown, equilateral triangle $ABC$ is rotated $75\degree$ clockwise about point $A$ to make equilateral triangle $ADE$ . Find the measure of angle (1) (marked at point $A$ between side $AB$ and side $AE$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

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

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1The rotation angle from AB to AD is 75 degrees

#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.

\angle BAD = 75^\circ

The amount you turn the shape is exactly the angle between a point and its rotated image.

2The rotated triangle keeps its 60-degree angle at A

#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.

\angle DAE = 60^\circ

Turning a shape does not stretch or bend it, so every angle stays the same size.

3Subtract to find angle 1

#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.

\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

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. This only needs Grade 4 angle-subtracting plus knowing a turn keeps every angle the same size!

Variant 2 easy

As shown, equilateral triangle $ABC$ is rotated $80\degree$ clockwise about point $A$ to make equilateral triangle $ADE$ . Find the measure of angle (1) (marked at point $A$ between side $AB$ and side $AE$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

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

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1The rotation angle from AB to AD is 80 degrees

#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.

\angle BAD = 80^\circ

The amount you turn the shape is exactly the angle between a point and its rotated image.

2The rotated triangle keeps its 60-degree angle at A

#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.

\angle DAE = 60^\circ

Turning a shape does not stretch or bend it, so every angle stays the same size.

3Subtract to find angle 1

#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.

\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

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. This only needs Grade 4 angle-subtracting plus knowing a turn keeps every angle the same size!

Variant 3 easy

As shown, equilateral triangle $ABC$ is rotated $90\degree$ clockwise about point $A$ to make equilateral triangle $ADE$ . Find the measure of angle (1) (marked at point $A$ between side $AB$ and side $AE$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

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

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1The rotation angle from AB to AD is 90 degrees

#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.

\angle BAD = 90^\circ

The amount you turn the shape is exactly the angle between a point and its rotated image.

2The rotated triangle keeps its 60-degree angle at A

#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.

\angle DAE = 60^\circ

Turning a shape does not stretch or bend it, so every angle stays the same size.

3Subtract to find angle 1

#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.

\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

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. This only needs Grade 4 angle-subtracting plus knowing a turn keeps every angle the same size!

Variant 4 medium

As shown, equilateral triangle $ABC$ is rotated $95\degree$ clockwise about point $A$ to make equilateral triangle $ADE$ . Find the measure of angle (1) (marked at point $A$ between side $AB$ and side $AE$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

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

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1The rotation angle from AB to AD is 95 degrees

#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.

\angle BAD = 95^\circ

The amount you turn the shape is exactly the angle between a point and its rotated image.

2The rotated triangle keeps its 60-degree angle at A

#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.

\angle DAE = 60^\circ

Turning a shape does not stretch or bend it, so every angle stays the same size.

3Subtract to find angle 1

#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.

\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

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. This only needs Grade 4 angle-subtracting plus knowing a turn keeps every angle the same size!

Variant 5 medium

As shown, equilateral triangle $ABC$ is rotated $100\degree$ clockwise about point $A$ to make equilateral triangle $ADE$ . Find the measure of angle (1) (marked at point $A$ between side $AB$ and side $AE$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

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

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1The rotation angle from AB to AD is 100 degrees

#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.

\angle BAD = 100^\circ

The amount you turn the shape is exactly the angle between a point and its rotated image.

2The rotated triangle keeps its 60-degree angle at A

#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.

\angle DAE = 60^\circ

Turning a shape does not stretch or bend it, so every angle stays the same size.

3Subtract to find angle 1

#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.

\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

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. This only needs Grade 4 angle-subtracting plus knowing a turn keeps every angle the same size!

Variant 6 medium

As shown, equilateral triangle $ABC$ is rotated $110\degree$ clockwise about point $A$ to make equilateral triangle $ADE$ . Find the measure of angle (1) (marked at point $A$ between side $AB$ and side $AE$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

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

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1The rotation angle from AB to AD is 110 degrees

#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.

\angle BAD = 110^\circ

The amount you turn the shape is exactly the angle between a point and its rotated image.

2The rotated triangle keeps its 60-degree angle at A

#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.

\angle DAE = 60^\circ

Turning a shape does not stretch or bend it, so every angle stays the same size.

3Subtract to find angle 1

#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.

\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

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. This only needs Grade 4 angle-subtracting plus knowing a turn keeps every angle the same size!

Variant 7 medium

As shown, equilateral triangle $ABC$ is rotated $120\degree$ clockwise about point $A$ to make equilateral triangle $ADE$ . Find the measure of angle (1) (marked at point $A$ between side $AB$ and side $AE$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

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

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1The rotation angle from AB to AD is 120 degrees

#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.

\angle BAD = 120^\circ

The amount you turn the shape is exactly the angle between a point and its rotated image.

2The rotated triangle keeps its 60-degree angle at A

#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.

\angle DAE = 60^\circ

Turning a shape does not stretch or bend it, so every angle stays the same size.

3Subtract to find angle 1

#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.

\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

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. This only needs Grade 4 angle-subtracting plus knowing a turn keeps every angle the same size!

Variant 8 hard

As shown, equilateral triangle $ABC$ is rotated $130\degree$ clockwise about point $A$ to make equilateral triangle $ADE$ . Find the measure of angle (1) (marked at point $A$ between side $AB$ and side $AE$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

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

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1The rotation angle from AB to AD is 130 degrees

#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.

\angle BAD = 130^\circ

The amount you turn the shape is exactly the angle between a point and its rotated image.

2The rotated triangle keeps its 60-degree angle at A

#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.

\angle DAE = 60^\circ

Turning a shape does not stretch or bend it, so every angle stays the same size.

3Subtract to find angle 1

#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.

\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

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. This only needs Grade 4 angle-subtracting plus knowing a turn keeps every angle the same size!

Variant 9 hard

As shown, equilateral triangle $ABC$ is rotated $150\degree$ clockwise about point $A$ to make equilateral triangle $ADE$ . Find the measure of angle (1) (marked at point $A$ between side $AB$ and side $AE$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

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

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1The rotation angle from AB to AD is 150 degrees

#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.

\angle BAD = 150^\circ

The amount you turn the shape is exactly the angle between a point and its rotated image.

2The rotated triangle keeps its 60-degree angle at A

#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.

\angle DAE = 60^\circ

Turning a shape does not stretch or bend it, so every angle stays the same size.

3Subtract to find angle 1

#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.

\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

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. This only needs Grade 4 angle-subtracting plus knowing a turn keeps every angle the same size!

Variant 10 hard

As shown, equilateral triangle $ABC$ is rotated $160\degree$ clockwise about point $A$ to make equilateral triangle $ADE$ . Find the measure of angle (1) (marked at point $A$ between side $AB$ and side $AE$ ).

Your answer

Show solution

1 · Understandwhat's really being asked

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

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1The rotation angle from AB to AD is 160 degrees

#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.

\angle BAD = 160^\circ

The amount you turn the shape is exactly the angle between a point and its rotated image.

2The rotated triangle keeps its 60-degree angle at A

#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.

\angle DAE = 60^\circ

Turning a shape does not stretch or bend it, so every angle stays the same size.

3Subtract to find angle 1

#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.

\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

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. This only needs Grade 4 angle-subtracting plus knowing a turn keeps every angle the same size!

Rotation preserves side lengths and angle measures · 10 practice problems

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