Question

How many triangles exist with the given angle measures?

80°, 50°, 50°
A.
No triangle exists with the given angle measures.

B.
More than one unique triangle exists with the given angle measures.

C.
Exactly one unique triangle exists with the given angle measures.

Answer 1

To determine how many triangles can be formed with the given angle measures of 80°, 50°, and 50°, we need to consider the properties of triangles.

1. **Sum of Angles**: The sum of the angles in any triangle must equal 180°. In this case:
\[
80° + 50° + 50° = 180°
\]
This condition is satisfied.

2. **Triangle Type**: The angles 80° and two angles of 50° indicate that this is an isosceles triangle (two angles are equal).

3. **Uniqueness**: For triangles that have a unique set of angles, the triangle is uniquely determined since angles define the shape of a triangle. In Euclidean geometry, if the angles are fixed, the triangle formed is unique regardless of size (it can be scaled).

Since we have confirmed that the angles indeed sum up to 180° and they form a valid triangle, we conclude:

- There is exactly ***one unique triangle*** that can be formed with the angles 80°, 50°, and 50°.

Therefore, the correct answer is:

**C. Exactly one unique triangle exists with the given angle measures.**