https://file.aiquickdraw.com/mathbot/file/172771508757265uebx45.jpg

The image contains a triangle with some known and unknown angles. Here's the breakdown of the problem:

1. One triangle is marked with angles \(54^\circ\), \(60^\circ\), and an unknown angle \(y^\circ\).
2. The other triangle adjacent to it shares the angle \(y^\circ\), and it has another known angle of \(65^\circ\), with the last unknown angle marked as \(x^\circ\).

### Steps to solve:
1. **For the first triangle** (with angles \(54^\circ\), \(60^\circ\), and \(y^\circ\)):
   The sum of the angles in any triangle is \(180^\circ\). So,
   \[
   54^\circ + 60^\circ + y^\circ = 180^\circ
   \]
   Solving for \(y^\circ\),
   \[
   y^\circ = 180^\circ - 54^\circ - 60^\circ = 66^\circ
   \]

2. **For the second triangle** (with angles \(y^\circ = 66^\circ\), \(65^\circ\), and \(x^\circ\)):
   Again, the sum of the angles in this triangle must also equal \(180^\circ\). Therefore,
   \[
   65^\circ + 66^\circ + x^\circ = 180^\circ
   \]
   Solving for \(x^\circ\),
   \[
   x^\circ = 180^\circ - 65^\circ - 66^\circ = 49^\circ
   \]

### Final answers:
- \(y = 66^\circ\)
- \(x = 49^\circ\)

Would you like a more detailed explanation, or do you have any further questions?

Here are 5 related questions:
1. How do we calculate the exterior angle of a triangle?
2. What is the sum of interior angles in a polygon with 5 sides?
3. Can the triangle inequality theorem be applied to angle measurements?
4. What happens to the sum of angles in non-Euclidean geometries?
5. How would you approach solving for unknown angles in a quadrilateral?

**Tip:** Always remember, the sum of the angles in any triangle will always be \(180^\circ\), regardless of its shape!

Find the unknown angle measures in a triangle where two angles are known (54° and 60°) and the third angle is shared with an adjacent triangle that has one known angle of 65°.

This tutorial explains how to solve for unknown angles in a triangle when two angles are known. We use the triangle angle sum theorem and solve for the shared angle between two triangles. Suitable for Grades 7-9 geometry students.

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