https://file.aiquickdraw.com/mathbot/file/1726725721226lmqjnwvt.jpg

The image shows a geometric problem where you are asked to find the values of \( x \), \( y \), and \( z \) based on the figure. Here's the breakdown of the relevant elements from the figure and given information:

### Information from the image:
- The triangle is labeled \( \triangle ABC \), with additional points \( D \) and \( E \) outside the triangle.
- \( \angle C \) is marked as \( 70^\circ \), and the angle adjacent to \( \angle C \) (on line \( CD \)) is labeled \( z \).
- \( \angle A \) is labeled \( x \).
- \( \angle B \) is labeled \( y \).
- Line segments \( AB \) and \( AC \) are marked as equal (suggesting \( \triangle ABC \) is isosceles with \( AB = AC \)).

### Steps to find the angles:

1. **Angle at \( C \)**: 
   - Since \( \angle C = 70^\circ \), and the line \( CD \) continues straight, the exterior angle \( z \) adjacent to \( C \) can be found using the linear pair relationship:
   \[
   \angle C + z = 180^\circ
   \]
   So,
   \[
   z = 180^\circ - 70^\circ = 110^\circ.
   \]

2. **Isosceles Triangle Property**:
   - Since \( \triangle ABC \) is isosceles with \( AB = AC \), the angles opposite these sides are equal:
   \[
   \angle A = \angle B = x = y.
   \]

3. **Sum of Angles in a Triangle**:
   - In any triangle, the sum of the interior angles is \( 180^\circ \). For \( \triangle ABC \):
   \[
   x + y + \angle C = 180^\circ.
   \]
   Substituting the values:
   \[
   x + x + 70^\circ = 180^\circ
   \]
   \[
   2x = 110^\circ \implies x = 55^\circ.
   \]
   Therefore, \( y = x = 55^\circ \).

### Final Answers:
- \( x = 55^\circ \),
- \( y = 55^\circ \),
- \( z = 110^\circ \).

Let me know if you'd like further clarification or additional steps! 

Here are some related questions to explore:
1. How would the solution change if \( AB \neq AC \)?
2. What is the sum of exterior angles of any triangle?
3. How can you prove the isosceles triangle property geometrically?
4. What would \( z \) be if \( \angle C \) were \( 80^\circ \)?
5. Can you find an alternate method to calculate \( z \)?

**Tip**: Always look for properties like equal sides or parallel lines in geometry to simplify angle calculations!

Find the values of x, y, and z in the given figure where triangle ABC is isosceles with AB = AC, angle C = 70°, and external angle z formed by line CD.

Learn how to find the values of x, y, and z in an isosceles triangle where AB = AC and angle C is 70°. This problem covers key geometry concepts like isosceles triangles and the linear pair theorem. Suitable for Grades 8-10 students.

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