1996 AJHSME Problems/Problem 24: Difference between revisions

Revision as of 07:39, 6 October 2024

Problem

The measure of angle $ABC$ is $50^\circ$ , $\overline{AD}$ bisects angle $BAC$ , and $\overline{DC}$ bisects angle $BCA$ . The measure of angle $ADC$ is

$[asy] pair A,B,C,D; A = (0,0); B = (9,10); C = (10,0); D = (6.66,3); dot(A); dot(B); dot(C); dot(D); draw(A--B--C--cycle); draw(A--D--C); label("$A$",A,SW); label("$B$",B,N); label("$C$",C,SE); label("$D$",D,N); label("$50^\circ $",(9.4,8.8),SW); [/asy]$

$\text{(A)}\ 90^\circ \qquad \text{(B)}\ 100^\circ \qquad \text{(C)}\ 115^\circ \qquad \text{(D)}\ 122.5^\circ \qquad \text{(E)}\ 125^\circ$

Solution

Let $\angle CAD = \angle BAD = x$ , and let $\angle ACD = \angle BCD = y$

From $\triangle ABC$ , we know that $50 + 2x + 2y = 180$ , leading to $x + y = 65$ .

From $\triangle ADC$ , we know that $x + y + \angle D = 180$ . Plugging in $x + y = 65$ , we get $\angle D = 180 - 65 = 115$ , which is answer $\boxed{C}$ .

Solution 2

Contruct $\overline{BE}$ through $D$ and intersects $\overline{AC}$ at point $E$

By Exterior Angle Theorem,

$\angle{ADE}$ $=$ $\angle{ABD} + \angle{BAD}$

Similarly,

$\angle{EDC} = \angle{DBC} + \angle{BCD}$

Thus,

$\angle{ADC} = \angle{ABC} + \angle{BAD} + \angle{BCD}$

Let $\angle{ADC} = x$

Because $\overline{AD}$ and $\overline{CD}$ are angle bisectors,

\begin{align*}

180^\circ - x = \angle{BAD} + \angle{BCD} \\

&= x - 50^\circ \\

\end{align*} (Error compiling LaTeX. Unknown error_msg)

$x = 115^\circ$

Thus, the answer is $\boxed{C}$

~ lovelearning999

1996 AJHSME (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
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All AJHSME/AMC 8 Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: Unable to save thumbnail to destination

@@ Line 46: / Line 46: @@
 Because <math>\overline{AD}</math> and <math>\overline{CD}</math> are angle bisectors,
-<math>180^\circ - x = \angle{BAD} + \angle{BCD}</math>
+<cmath>\begin{align*}
-        <math>= x - 50^\circ</math>
+^\circ - x = \angle{BAD} + \angle{BCD} \\
+&= x - 50^\circ \\
+\end{align*}</cmath>
 <math>x = 115^\circ</math>

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1996 AJHSME Problems/Problem 24: Difference between revisions

Revision as of 07:39, 6 October 2024

Contents

Problem

Solution

Solution 2

See Also