In the figure above, and intersect at point P , , and . What is the measure, in degrees, of

Question

In the figure above, and intersect at point P , , and . What is the measure, in degrees, of ? (Disregard the degree symbol when gridding your answer.)

Accepted Answer

The correct answer is 30. It is given that the measure of $angle Q P R$ is $60 degrees$ . Angle MPR and $angle Q P R$ are collinear and therefore are supplementary angles. This means that the sum of the two angle measures is $180 degrees$ , and so the measure of $angle M P R$ is $120 degrees$ . The sum of the angles in a triangle is $180 degrees$ . Subtracting the measure of $angle M P R$ from $180 degrees$ yields the sum of the other angles in the triangle MPR. Since $180 minus 120, equals 60$ , the sum of the measures of $angle Q M R$ and $angle N R M$ is $60 degrees$ . It is given that $the length of side M P equals the length of side P R$ , so it follows that triangle MPR is isosceles. Therefore $angle Q M R$ and $angle N R M$ must be congruent. Since the sum of the measure of these two angles is $60 degrees$ , it follows that the measure of each angle is $30 degrees$ .

An alternate approach would be to use the exterior angle theorem, noting that the measure of $angle Q P R$ is equal to the sum of the measures of $angle Q M R$ and $angle N R M$ . Since both angles are equal, each of them has a measure of $30 degrees$ .

Question in-the

In the figure above, and intersect at point P , , and . What is the

More Lines, Angles, and Triangles practice questions