1954 AMC 12 Problem 37

Attempt Problem 37 of the 1954 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1954 AMC 12 solutions, or check the answer key.

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37.

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.

Problem 37 in other years