Question

in the diagram below \\( \angle pqr = 132 ^ { \circ } \\) and \\( \angle prs = 163 ^ { \circ } \\). find the measure of \\( \angle qpr \\).
warm up lesson 5.1
\\( 180 ^ { \circ } \\)
\\( 295 ^ { \circ } \\)
\\( 17 ^ { \circ } \\)
\\( 31 ^ { \circ } \\)

Explanation:

Step1: Find the measure of ∠PRQ

Since ∠PRS and ∠PRQ are supplementary angles (they form a linear pair), we know that ∠PRS+∠PRQ = 180°.
Given ∠PRS = 163°, then ∠PRQ=180° - 163°=17°.

Step2: Use the exterior - angle theorem

The exterior - angle theorem states that in a triangle, an exterior angle is equal to the sum of the two non - adjacent interior angles.
In △PQR, ∠PQR is an interior angle, ∠PRQ is an interior angle, and ∠QPR is the angle we want to find. Also, ∠PQR (132°) is an exterior angle for the "small" triangle formed with ∠PRQ (17°) and ∠QPR.
Let ∠QPR=x. Then, using the exterior - angle theorem: ∠PQR=∠PRQ + ∠QPR.
Substitute the known values: 132°=17°+x.
Solve for x: x = 132°-17° = 31°.

Answer:

31°