Triangles ABC and ADC are isosceles with AB=BC and AD=DC. Point D is inside △ ABC, ∠ ABC=40°, and ∠ ADC=140°. What is the degree measure of ∠ BAD?

2007 AMC 12A Problem 6

Attempt Problem 6 of the 2007 AMC 12A 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 2007 AMC 12A solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

Triangles

ABC

and

ADC

are isosceles with

AB=BC

and

AD=DC.

Point

D

is inside

\triangle ABC,

\angle ABC=40^\circ,

and

\angle ADC=140^\circ.

What is the degree measure of

\angle BAD?

20

30

40

50

60

In each isosceles triangle the two base angles are equal

\angle BAC=\tfrac12(180^\circ-40^\circ)

and

\angle DAC=\tfrac12(180^\circ-140^\circ)

; subtract

Since

\triangle ABC

is isosceles,

\angle BAC=\tfrac12(180^\circ-\angle ABC)

=70^\circ.

Since

\triangle ADC

is isosceles,

\angle DAC=\tfrac12(180^\circ-\angle ADC)

=20^\circ.

Therefore

\angle BAD=\angle BAC-\angle DAC

=70^\circ-20^\circ

=50^\circ.

Thus, the correct answer is D.