Question

1. to create this diagram, triangle abc was translated so that a goes to c. then, triangle abc was translated so that a goes to b. the measure of angle a is 45° and the measure of angle d is 63°. a. identify at least two pairs of congruent angles in the figure and explain how you know they are congruent. b. what is the measure of angle cbe? explain how you know. c. name a triangle congruent to triangle cbe. explain how you know.

Explanation:

Step1: Identify congruent - angles (a)

Since translation preserves angle - measure, $\angle A=\angle CFE$ and $\angle A=\angle BCE$ because translation is a rigid motion that does not change the shape or size of the figure, including angle - measures.

Step2: Find measure of $\angle CBE$ (b)

In $\triangle ABD$, we know $\angle A = 45^{\circ}$ and $\angle D=63^{\circ}$. By the angle - sum property of a triangle ($\angle A+\angle ABD+\angle D = 180^{\circ}$), $\angle ABD=180^{\circ}-\angle A - \angle D=180^{\circ}-45^{\circ}-63^{\circ}=72^{\circ}$. $\angle CBE$ and $\angle ABD$ are the same angle, so $\angle CBE = 72^{\circ}$.

Step3: Find congruent triangle (c)

$\triangle CBE\cong\triangle ABC$. Because of the translations described, the side - lengths and angle - measures of $\triangle ABC$ are preserved in $\triangle CBE$. Translation is a rigid transformation that ensures congruence of the pre - image and the image.

Answer:

a. $\angle A\cong\angle CFE$ and $\angle A\cong\angle BCE$ due to translation preserving angle - measure.
b. $72^{\circ}$, found using the angle - sum property of a triangle in $\triangle ABD$.
c. $\triangle ABC$, because translation is a rigid transformation that preserves side - lengths and angle - measures.