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In the figure above, bar(MQ) and bar(NR)...


In the figure above, `bar(MQ)` and `bar(NR)` intersect at point P, NP = QP, and MP = PR. What is the measure, in degrees, of `angleQMR` ? (Disregard the degree symbol when gridding your answer.)

Text Solution

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It is given that the measure of `angle`QPR is `60^@`. Angle MPR and `angle`QPR are collinear and therefore are supplementary angles. This means that the sum of the two angle measures is `180^@`, and so the measure of `angle`MPR is `120^@`. The sum of the angles in a triangle is `180^@`. Subtracting the measure of `angle`MPR from `180^@` yields the sum of the other angles in the triangle MPR. Since 180 − 120 = 60, the sum of the measures of `angle`QMR and `angle`NRM is `60^@`. It is given that MP = PR, so it follows that triangle MPR is isosceles. Therefore `angle`QMR and `angle`NRM must be congruent. Since the sum of the measure of these two angles is `60^@`, it follows that the measure of each angle is `30^@`.
An alternate approach would be to use the exterior angle theorem, noting that the measure of `angle`QPR is equal to the sum of the measures of `angle`QMR and `angle`NRM. Since both angles are equal, each of them has a measure of `30^@`.
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