1997 AMC 8 Problem 12

Attempt Problem 12 of the 1997 AMC 8 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 1997 AMC 8 solutions, or check the answer key.

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12.

∠1+∠2=180∘\angle 1 + \angle 2 = 180^\circ ∠3=∠4\angle 3 = \angle 4 Find ∠4\angle 4
20∘20^\circ
25∘25^\circ
30∘30^\circ
35∘35^\circ
40∘40^\circ
Answer: D
Concepts:angle chasingangle sum
Small hint:
Use the supplementary condition and then angle 33 equals angle 44
Big hint:
First find angle 11 from the left triangle
Solution:
We have that ∠1=180∘−70∘−40∘=70∘ \angle 1 = 180^{\circ} - 70^{\circ} - 40^{\circ} = 70^{\circ} using the fact that the interior angles of a triangle add to 180∘.180^{\circ}.
This tells us that ∠2=180∘−∠1=110∘ \angle 2 = 180^{\circ} - \angle 1 = 110^{\circ} since ∠1\angle 1 and ∠2\angle 2 are supplementary.
Finally, ∠3+∠4+110∘=180∘⇒ \angle 3 + \angle 4 + 110^{\circ} = 180^{\circ} \Rightarrow2∠4=70∘⇒∠4=35∘. 2 \angle 4 = 70^{\circ} \Rightarrow \angle 4 = 35^{\circ}.
Thus, D is the correct answer.
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