Question

In triangle, P, Q, R, comma△PQR, start overline, Q, R, end overline, \cong, start overline, P, Q, end overline

QR
​
≅
PQ
​
and m, angle, Q, equals, 84, degrees, .m∠Q=84
∘
. Find m, angle, P, .m∠P

Answer 1

In triangle PQR, since QR ≅ PQ and m∠Q = 84°, triangle PQR is isosceles with m∠P = m∠R. The sum of angles in a triangle is 180°, which allows us to calculate m∠P as 48°.

Step-by-step explanation:

The student has stated that in triangle PQR, the side lengths of QR are congruent to PQ and that the measure of angle Q is 84 degrees. To find the measure of angle P, we must understand that since two sides of the triangle are congruent, triangle PQR is an isosceles triangle. In an isosceles triangle, the angles opposite the congruent sides are equal. Therefore, angles P and R are equal.

Since the sum of angles in any triangle is 180 degrees, we can set up the equation m∠P + m∠Q + m∠R = 180°. We know that m∠Q is 84° and that m∠P = m∠R. This gives us 2m∠P + 84° = 180°. Subtracting 84° from both sides, we get 2m∠P = 96°. Dividing by 2, we find that m∠P = 48°.