https://file.aiquickdraw.com/mathbot/file/17254605903465nswpfer.jpg

To solve this problem, let's focus on the triangle provided. The angles given are:

- Angle 1: 34°
- Angle 2: 59°
- Angle 3 (unknown, marked as z°)
- Angle 4 (unknown, marked as y°)

Since it's a triangle, the sum of the internal angles of any triangle is always 180°. 

### Step 1: Calculate Angle \( z \)
From the image, the triangle has angles 34°, 59°, and \( z \). We know that:
\[
34° + 59° + z° = 180°
\]
\[
z° = 180° - 34° - 59°
\]
\[
z° = 87°
\]

Thus, the measure of angle \( z \) is 87°.

### Step 2: Calculate Angle \( y \)
From the diagram, it appears there is an external angle formed with one of the sides of the triangle. We know that the external angle is equal to the sum of the two opposite interior angles. In this case, the exterior angle \( y \) should be equal to the sum of 34° and 59°:
\[
y° = 34° + 59°
\]
\[
y° = 93°
\]

Thus, the measure of angle \( y \) is 93°.

### Final Answer:
- \( z = 87^\circ \)
- \( y = 93^\circ \)

Would you like further details on any of the steps? 

Here are 5 related questions for deeper understanding:

1. How do we use the triangle sum theorem to find unknown angles?
2. What is the relationship between interior and exterior angles of a triangle?
3. Can you prove that the sum of the angles in any triangle always equals 180°?
4. How does the classification of triangles change based on angle measures?
5. What are some real-life applications of triangle angle relationships?

**Tip:** Always verify the sum of the angles when solving for unknown angles in a triangle to avoid calculation errors.

Solve the problem of finding unknown angles in a triangle using the Triangle Sum Theorem and the Exterior Angle Theorem. This solution includes key geometry concepts and step-by-step calculations. Suitable for students in Grades 6-8.

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