1958 AMC 12 Problem 29

Attempt Problem 29 of the 1958 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 1958 AMC 12 solutions, or check the answer key.

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29.

In a general triangle $ADE$ (as shown), lines $EB$ and $EC$ are drawn. Which of the following angle relations is true?

$x+z=a+b$

$y+z=a+b$

$m+x=w+n$

$x+z+n=w+c+m$

$x+y+n=a+b+m$

Write the angle sum in triangle $AEC$

Write the angle sum in triangle $BED$ , then compare the two equations

Triangle $AEC$ gives $x+y+w+n=180^\circ.$ Triangle $BED$ gives $m+a+b+w=180^\circ.$ Equating the left sides and cancelling $w$ yields $x+y+n=a+b+m.$

Thus, the correct answer is E.