Question

Given triangle $PQR$ with $\overline{RS}$ bisecting $\angle R,$ $\overline{PQ}$ extended to $D,$ and $\angle n$ a right angle, then:

$\angle m=\dfrac12(\angle p-\angle q)$

$\angle m=\dfrac12(\angle p+\angle q)$

$\angle d=\dfrac12(\angle q+\angle p)$

$\angle d=\dfrac12\angle m$

none of these is correct

Answer: B

Concepts:angle bisector angle chasing angle sum

Difficulty rating: 1780

Small Hint:

The bisected angle at $R$ measures $180^\circ-\angle p-\angle q$ before it is halved

Big Hint:

Since the transversal is perpendicular to the angle bisector, $\angle m$ complements half of $\angle R$

Solution:

The angle at $R$ is $180^\circ-\angle p-\angle q.$ Because $\overline{RS}$ bisects this angle, the angle between $\overline{PR}$ and $\overline{RS}$ is $90^\circ-\frac12(\angle p+\angle q).$ The line forming $\angle m$ is perpendicular to $\overline{RS},$ so $\angle m=\frac12(\angle p+\angle q).$

Thus, the correct answer is B.

1954 AMC 12 Problem 37

Answer: B

Difficulty rating: 1780

Small Hint:

Big Hint:

Solution:

Problem 37 in Other Years