Question

triangle \\( \triangle a b c \\) is rotated \\( - 120 ^ { \circ } \\) about point \\( p \\) to create \\( \triangle a ^ { \prime } b ^ { \prime } c ^ { \prime } \\).
what is the measure of \\( \angle a ^ { \prime } \\)?

Explanation:

Step1: Recall the property of rotation

Rotation is a rigid transformation. Rigid transformations (such as rotation) preserve the shape and size of the figure. So, \(\triangle ABC\cong\triangle A'B'C'\).

Step2: Use the angle - congruence property of congruent triangles

If \(\triangle ABC\cong\triangle A'B'C'\), then the corresponding angles are congruent. That is, \(\angle A=\angle A'\).

Step3: Calculate the measure of \(\angle A\)

We know that the sum of the interior angles of a triangle is \(180^{\circ}\). In \(\triangle ABC\), if we assume \(\angle C = 105^{\circ}\) (corresponding to \(\angle C'\) in the rotated triangle) and using the angle - sum formula for a triangle \(S=\angle A+\angle B+\angle C = 180^{\circ}\). Let's assume we use the fact that for \(\triangle A'B'C'\) (since \(\triangle ABC\cong\triangle A'B'C'\)), \(\angle A'+\angle B'+\angle C'=180^{\circ}\). But more simply, since \(\angle A\) and \(\angle A'\) are corresponding angles of congruent triangles.
We know that \(\angle A = 35^{\circ}\) (because \(180-(105 + 40)=35\)).

Answer:

\(35\)