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 $\boxed{\textbf{(D)}\    165}$.

Video Solution

https://youtu.be/FDgcLW4frg8?t=902

~ pi_is_3.14

See Also

2009 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 18
Followed by
Problem 20
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All AJHSME/AMC 8 Problems and Solutions

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