https://file.aiquickdraw.com/mathbot/file/1726682734754fdb2goyf.jpg

Let's go through the problem step by step, based on the given image.

### a. Identify at least two pairs of congruent angles in the figure and explain how you know they are congruent.

1. **∠A = ∠C**: Since triangle ABC was translated, the corresponding angles remain the same. Therefore, angle A and angle C are congruent.
   
2. **∠B = ∠D**: Similarly, triangle ABC was again translated so that point A goes to B. Because of this translation, angle B and angle D are congruent as well.

### b. What is the measure of angle ∠CBE? Explain how you know.

- From the given data, angle ∠D is 63°, and since triangle CBE is part of a larger triangle ABD, we can use the fact that the sum of angles in a triangle is always 180°. In triangle ABD:

\[
\text{Sum of angles} = 180^\circ
\]

- Given that ∠A = 45° and ∠D = 63°, we can calculate ∠B:

\[
\text{∠B = 180° - (45° + 63°) = 72°}
\]

Since ∠B = 72° and ∠CBE is a part of the same angle, the measure of ∠CBE is **72°**.

### c. Name a triangle congruent to triangle CBE. Explain how you know.

- Triangle **ABD** is congruent to triangle **CBE**. This is because triangles ABD and CBE share the same corresponding angles (∠A = ∠C, ∠B = ∠E, and ∠D = ∠B). These triangles were created by translations, preserving congruency.

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Would you like further clarification or details on any of these answers?

Here are 5 related questions to deepen your understanding:

1. How does translation affect the angles and sides of a triangle?
2. Can you identify any other pairs of congruent angles or sides in the diagram?
3. What other transformations, besides translation, preserve congruency?
4. How would you prove that two triangles are congruent using the SSS, SAS, or ASA rules?
5. How can you calculate missing angles when given a combination of triangle and exterior angle information?

**Tip:** When translating a shape, all angles and side lengths remain the same, ensuring congruent figures!

Identify at least two pairs of congruent angles, find the measure of angle CBE, and name a triangle congruent to triangle CBE.

Understand how to identify congruent angles in translated triangles, calculate the measure of angle CBE, and find congruent triangles in this Grade 8 math problem. The problem uses the Triangle Angle Sum Theorem and explores translations.

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