Let's analyze the figure and solve for the unknown angles step-by-step.

Given Information

1. $\angle A = 70^\circ$
2. $\angle B = 65^\circ$
3. $\angle C = 65^\circ$

Approach and Reasoning

We'll use the properties of triangles and parallel lines (alternate interior angles, vertically opposite angles, etc.) to determine the unknown angles.

Step 1: Finding $\angle E$

The triangle at the top (formed by $\angle A$, $\angle B$, and $\angle E$) is a triangle, and we know: $\angle A + \angle B + \angle E = 180^\circ$ Substituting the values: $70^\circ + 65^\circ + \angle E = 180^\circ$ $\angle E = 180^\circ - 135^\circ = 45^\circ$ So, $\angle E = 45^\circ$.

Step 2: Finding $\angle D$

The middle triangle, which includes $\angle A$, $\angle D$, and $\angle C$, is also a triangle. Thus, $\angle A + \angle C + \angle D = 180^\circ$ Substitute the known values: $70^\circ + 65^\circ + \angle D = 180^\circ$ $\angle D = 180^\circ - 135^\circ = 45^\circ$ So, $\angle D = 45^\circ$.

Final Results

• $\angle E = 45^\circ$
• $\angle D = 45^\circ$

These solutions are based on the properties of triangles (the sum of the angles in a triangle is $180^\circ$) and the values given.

Would you like a detailed breakdown of each step or further explanation?

Here are five related questions for practice:

1. What is the value of the third angle in a triangle if the two given angles are $80^\circ$ and $60^\circ$?
2. How can we find unknown angles when two parallel lines are cut by a transversal?
3. Explain how alternate interior angles work in parallel lines.
4. What is the sum of angles in any quadrilateral?
5. How do vertically opposite angles help in finding unknown angles?

Tip: Remember that the sum of angles in any triangle is always $180^\circ$.