2000 AMC 8 Problem 13

Question

In triangle

CAT,

we have

\angle ACT =\angle ATC

and

\angle CAT = 36^\circ.

If

\overline{TR}

bisects

\angle ATC,

then

\angle CRT =

16^\circ

51^\circ

72^\circ

90^\circ

108^\circ

First find the two equal base angles of

\triangle CAT

Use the angle bisector at

T

We get

\begin{aligned} \angle CAT + \angle ATC \\ \quad {}+ \angle ACT &= 180 \\ 36 + 2 \cdot \angle ATC &= 180 \\ \angle ATC &= 72^{\circ}. \end{aligned}

Due to bisection, we also know that

\angle RTC = 72 \div 2 = 36^{\circ}.

Finally, we see that

\begin{aligned} \angle RTC + \angle TCR \\ \quad {}+ \angle CRT &= 180 \\ 36 + 72 + \angle CRT &= 180 \\ \angle CRT &= 72^{\circ}. \end{aligned}

Thus, C is the correct answer.

Problem 13 in other years