1996 AJHSME Problems/Problem 24

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,

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

       $= x - 50^\circ$

$x = 115^\circ$

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

~ lovelearning999

See Also

1996 AJHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 23
Followed by
Problem 25
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All AJHSME/AMC 8 Problems and Solutions

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