2009 AMC 8 Problems/Problem 19

Problem

Two angles of an isosceles triangle measure $70^\circ$ and $x^\circ$ . What is the sum of the three possible values of $x$ ?

$\textbf{(A)}\ 95 \qquad \textbf{(B)}\ 125 \qquad \textbf{(C)}\ 140 \qquad \textbf{(D)}\ 165 \qquad \textbf{(E)}\ 180$

Solution

There are 3 cases: where $x^\circ$ is a base angle with the $70^\circ$ as the other angle, where $x^\circ$ is a base angle with $70^\circ$ as the vertex angle, and where $x^\circ$ is the vertex angle with $70^\circ$ as a base angle.

Case 1: $x^\circ$ is a base angle with the $70^\circ$ as the other angle: Here, $x=70$ , since base angles are congruent.

Case 2: $x^\circ$ is a base angle with $70^\circ$ as the vertex angle: Here, the 2 base angles are both $x^\circ$ , so we can use the equation $2x+70=180$ , which simplifies to $x=55$ .

Case 3: $x^\circ$ is the vertex angle with $70^\circ$ as a base angle: Here, both base angles are $70^\circ$ , since base angles are congruent. Thus, we can use the equation $x+140=180$ , which simplifies to $x=40$ .

Adding up all the cases, we get $70+55+40=165$ , so the answer is $\textbf{(D)}\ 165$ .

2009 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 18	Followed by Problem 20
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2009 AMC 8 Problems/Problem 19

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