1. **Stating the problem:** We have a triangle ABC with angles at B and C given as 20° and 30° respectively. Points D and E lie on sides AC and AB, and angles at D and E are labeled as $y$ and $z$. We want to find the values of $y$ and $z$. 2. **Using triangle angle sum rule:** The sum of angles in any triangle is 180°. For triangle ABC, $$\angle A + \angle B + \angle C = 180^\circ$$ Given $\angle B = 20^\circ$ and $\angle C = 30^\circ$, we find $\angle A$: $$\angle A = 180^\circ - 20^\circ - 30^\circ = 130^\circ$$ 3. **Analyzing triangles ABD and ACE:** Since D lies on AC and E lies on AB, triangles ABD and ACE share angles with ABC. - Triangle ABD has angle at B = 20° (given). - Triangle ACE has angle at C = 30° (given). 4. **Using the given hints:** The problem states "centerABD = 20" and "ACE=30" which likely means angles at B in ABD and at C in ACE are 20° and 30° respectively, consistent with the main triangle. 5. **Finding $y$ and $z$:** Since $y$ is angle at D on AC and $z$ is angle at E on AB, and given the configuration, these angles correspond to the angles opposite to the known angles in the smaller triangles. 6. **Using the fact that angles on a straight line sum to 180°:** At point D on AC, angles $y$ and the adjacent angle from triangle ABC sum to 180°. Similarly for point E on AB, angles $z$ and the adjacent angle sum to 180°. 7. **Calculating $y$ and $z$:** Since $\angle A = 130^\circ$, and points D and E lie on sides AC and AB respectively, the angles $y$ and $z$ are complementary to the angles at B and C: $$y = 180^\circ - 130^\circ = 50^\circ$$ $$z = 180^\circ - 130^\circ = 50^\circ$$ **Final answers:** $$y = 50^\circ$$ $$z = 50^\circ$$