Question

In triangle pqr, if measure of angle r is 90 degrees, measure of angle pqr is 75 degrees, and measure of angle mqr is 60 degrees, what is the measure of angle mpq?

1) 15 degrees
2) 30 degrees
3) 45 degrees
4) 60 degrees

Answer 1

The measure of angle MPQ in triangle PQR, with the given angles, is 60 degrees. This is found by first determining the measure of angle QPR using the sum of angles in a triangle and then calculating angle MPQ by considering the supplementary angles along line MP.

Step-by-step explanation:

To find the measure of angle MPQ in triangle PQR, where angle R is 90 degrees, angle PQR is 75 degrees, and angle MQR is 60 degrees, we need to use the properties of angles in a triangle.

Since angle R is 90 degrees, we know that triangle PQR is a right-angled triangle at R. The sum of angles in any triangle is 180 degrees. Therefore, the measure of angle QPR is:

180 degrees - 90 degrees (angle R) - 75 degrees (angle PQR) = 15 degrees.

Now, considering triangle MQP, angle MQP and angle MQR form a straight line along QR, so they should add up to 180 degrees. Since angle MQR is 60 degrees, the measure of angle MQP is:

180 degrees - 60 degrees = 120 degrees.

Finally, angle MQP and angle MPQ add up to 180 degrees because they are supplementary angles as they form a straight line along MP. Therefore, the measure of angle MPQ is:

180 degrees - 120 degrees (angle MQP) = 60 degrees.

Hence, the correct answer for the measure of angle MPQ is 60 degrees (Option 4).