2008 AMC 10B Problems/Problem 24
Problem
Quadrilateral has
, angle
and angle
. What is the measure of angle
?
Solution
Draw the angle bisectors of the angles and
. These two bisectors obviously intersect. Let their intersection be
.
We will now prove that
lies on the segment
.
Note that the triangles and
are equal, as they share the side
, and we have
and
.
Also note that for similar reasons the triangles and
are equal.
Now we can compute their inner angles. is the bisector of the angle
, hence
, and thus also
.
is the bisector of the angle
, hence
, and thus also
.
It follows that . Thus the angle
has
, and hence
does indeed lie on
. Then obviously
.
See also
2008 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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All AMC 10 Problems and Solutions |