Rotation preserves side lengths and angle measures

4.MD.C.7 · take · grade 4

Archetype: Transformations Preserve Measures · step in a 8-type progression

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 $\textcircled{1}$ ).

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 90 degrees and where AE lands. Then I split the 90-degree turn into the part covered by the equilateral triangle's 60-degree angle 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.

\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.

\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.

\angle 1 = \angle BAD - \angle DAE = 90^\circ - 60^\circ = 30^\circ

Angle measure is additive, so the leftover piece of the 90-degree turn after the 60-degree angle is the answer.

Answer: 30 degrees

Review

Angle 1 = 30 degrees is acute and smaller than both the 90-degree rotation and the 60-degree triangle angle, which fits because it is the small leftover sliver between AE and AB. 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 by 90 degrees about A, and measure angle 1 with a protractor to confirm 30 degrees.

Standards · min grade 4

4.MD.C.5 Recognize angles as geometric shapes formed when two rays share an endpoint — Treating the 90-degree 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 60-degree angle, preserved under rotation.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting the 60-degree angle from the 90-degree rotation to find angle 1.

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