Difference between revisions of "2009 AMC 8 Problems/Problem 19"
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− | == | + | ==Problem== |
Two angles of an isosceles triangle measure <math> 70^\circ</math> and <math> x^\circ</math>. What is the sum of the three possible values of <math> x</math>? | Two angles of an isosceles triangle measure <math> 70^\circ</math> and <math> x^\circ</math>. What is the sum of the three possible values of <math> x</math>? | ||
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\textbf{(D)}\ 165 \qquad | \textbf{(D)}\ 165 \qquad | ||
\textbf{(E)}\ 180</math> | \textbf{(E)}\ 180</math> | ||
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+ | |||
+ | ==Solution== | ||
+ | |||
+ | There are 3 cases: where <math> x^\circ</math> is a base angle with the <math> 70^\circ</math> as the other angle, where <math> x^\circ</math> is a base angle with <math> 70^\circ</math> as the vertex angle, and where <math> x^\circ</math> is the vertex angle with <math> 70^\circ</math> as a base angle. | ||
+ | |||
+ | Case 1: <math> x^\circ</math> is a base angle with the <math> 70^\circ</math> as the other angle: | ||
+ | Here, <math> x=70</math>, since base angles are congruent. | ||
+ | |||
+ | Case 2: <math> x^\circ</math> is a base angle with <math> 70^\circ</math> as the vertex angle: | ||
+ | Here, the 2 base angles are both <math> x^\circ</math>, so we can use the equation <math> 2x+70=180</math>, which simplifies to <math> x=55</math>. | ||
+ | |||
+ | Case 3: <math> x^\circ</math> is the vertex angle with <math> 70^\circ</math> as a base angle: | ||
+ | Here, both base angles are <math> 70^\circ</math>, since base angles are congruent. Thus, we can use the equation <math> x+140=180</math>, which simplifies to <math> x=40</math>. | ||
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+ | Adding up all the cases, we get <math>70+55+40=165</math>, so the answer is <math> \textbf{(D)}\ 165</math>. | ||
==See Also== | ==See Also== | ||
{{AMC8 box|year=2009|num-b=18|num-a=20}} | {{AMC8 box|year=2009|num-b=18|num-a=20}} |
Revision as of 17:17, 12 November 2012
Problem
Two angles of an isosceles triangle measure and
. What is the sum of the three possible values of
?
Solution
There are 3 cases: where is a base angle with the
as the other angle, where
is a base angle with
as the vertex angle, and where
is the vertex angle with
as a base angle.
Case 1: is a base angle with the
as the other angle:
Here,
, since base angles are congruent.
Case 2: is a base angle with
as the vertex angle:
Here, the 2 base angles are both
, so we can use the equation
, which simplifies to
.
Case 3: is the vertex angle with
as a base angle:
Here, both base angles are
, since base angles are congruent. Thus, we can use the equation
, which simplifies to
.
Adding up all the cases, we get , so the answer is
.
See Also
2009 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |