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.