In \triangle PQR, the measure of \angle P is 50^\circ. The angle bisectors of \angle PQR a… — ACT

In $\triangle PQR$ , the measure of $\angle P$ is $50^\circ$ . The angle bisectors of $\angle PQR$ and $\angle PRQ$ intersect at point $I$ inside the triangle. What is the measure of $\angle QIR$ ?

A
$65^\circ$
B
$100^\circ$
$115^\circ$ Answer
D
$125^\circ$
E
$130^\circ$

Answer

115 degrees

To find the measure of the angle at the intersection, we first use the Triangle Angle-Sum Theorem on the larger triangle. The sum of the interior angles in the larger triangle is 180 degrees. Since the angle at one vertex is 50 degrees, the sum of the other two angles must be 130 degrees. The angle bisectors divide these two angles in half, meaning the sum of the two half-angles in the smaller triangle is half of 130 degrees, which is 65 degrees. Applying the Triangle Angle-Sum Theorem to the smaller triangle, the sum of its interior angles is also 180 degrees. Subtracting the sum of the two half-angles (65 degrees) from 180 degrees gives 115 degrees for the angle at the intersection.

Step-by-Step Solution

Find the sum of the remaining interior angles of the triangle.

The sum of the angles at Q and R is 130 degrees.

The sum of all interior angles in any triangle is always 180 degrees, and the angle at P is given as 50 degrees.

Determine the sum of the bisected angles in the smaller triangle.

The sum of the half-angles is 65 degrees.

Since the lines QI and RI are angle bisectors, the sum of the interior angles of the smaller triangle at vertices Q and R is half the sum of the angles at Q and R of the larger triangle.

Calculate the measure of the angle at the intersection point.

The angle at the intersection point is 115 degrees.

The sum of the interior angles in the smaller triangle is also 180 degrees, so the unknown angle is found by subtracting the sum of the two half-angles from 180 degrees.

Key Concept

The Triangle Angle-Sum Theorem states that the sum of the measures of the interior angles of a triangle is always 180 degrees. Angle bisectors divide an angle into two equal parts.

Alternative Method

Alternatively, one can assign specific values to the angles that satisfy the given conditions. Since the sum of the angles in the large triangle must be 180 degrees and the angle at vertex P is 50 degrees, the sum of the other two angles must be 130 degrees. If we assume the triangle is isosceles with the two unknown angles being equal, each of those angles is 65 degrees. Their bisectors would each form an angle of 32.5 degrees with the base. In the smaller triangle, the sum of these two bisected angles is 65 degrees, which leaves 115 degrees for the angle at the intersection point.

Estimated Time:1m 15s

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