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Find the measure of ∠*B* and** ∠*E* in the similar triangles.

To solve for the angles \( \angle B \) and \( \angle E \) in the similar triangles shown in the image, we will use the properties of similar triangles and the fact that the sum of the angles in a triangle is \( 180^\circ \).

### Step 1: Identify the angles
- Given: \( \angle A = 90^\circ \) (since \( \triangle ABE \) is a right triangle).
- \( \angle C = 53^\circ \).

### Step 2: Use the properties of similar triangles
- Triangles \( \triangle ABC \), \( \triangle ADE \), and \( \triangle EBC \) are all similar by the AA (Angle-Angle) similarity criterion.

### Step 3: Determine \( \angle B \)
- In \( \triangle ABC \), since \( \angle C = 53^\circ \) and \( \angle A = 90^\circ \), the third angle \( \angle B \) can be calculated as:
  \[
  \angle B = 180^\circ - \angle A - \angle C = 180^\circ - 90^\circ - 53^\circ = 37^\circ
  \]

### Step 4: Determine \( \angle E \)
- Since \( \triangle ADE \) is similar to \( \triangle ABC \), \( \angle E \) corresponds to \( \angle C \) in \( \triangle ABC \).
  - Therefore, \( \angle E = \angle C = 53^\circ \).

### Final Answer:
- \( \angle B = 37^\circ \)
- \( \angle E = 53^\circ \)

Would you like more details or have any further questions? 

Here are 5 related questions to expand your understanding:

1. How would you verify that the triangles are indeed similar?
2. What would be the lengths of sides \( AB \), \( AC \), and \( BC \) if given one of the side lengths?
3. How would you find \( \angle D \) in the triangle \( \triangle ADE \)?
4. Can you calculate the height of point \( E \) above \( AB \) if given the side lengths?
5. How does the sum of angles property apply to other polygons?

**Tip:** In similar triangles, corresponding angles are equal, and the ratios of corresponding sides are proportional.

Find the measure of ∠B and ∠E in the similar triangles.

Learn how to find the measures of angles ∠B and ∠E in similar triangles using the angle sum property and AA similarity criterion. This geometry problem is suitable for students in Grades 8-10.

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