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!