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smc

geometry senior

Problem

Given

△ P QR

with

\overline{R S}

bisecting

∠ R

,

P Q

extended to

D

and

∠ n

a right angle, then:

problem

(A)

∠ m = \frac{1}{2} (∠ p - ∠ q)

(B)

∠ m = \frac{1}{2} (∠ p + ∠ q)

(C)

∠ d = \frac{1}{2} (∠ q + ∠ p)

(D)

∠ d = \frac{1}{2} ∠ m

Solution

Let

∠ P R S

be

θ

.

p + q + 2 θ = 180

m + θ + 90 = 180 ⟹ m + θ = 90 ⟹ 2 m + 2 θ = 180

p + q + 2 θ = 2 m + 2 θ ⟹ \frac{p + q}{2} = m ⟹ ∠ m = \frac{1}{2} (∠ p + ∠ q)

Final answer

B