Super-Hard SAT Geometry Question 23 | Free SAT Prep

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Question 23·25 Super-Hard SAT Geometry and Trigonometry Questions·Geometry and Trigonometry

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Points $A$ , $C$ , and $D$ are collinear, with $C$ between $A$ and $D$ . The lengths of $\overline{AB}$ and $\overline{BD}$ are equal, and the lengths of $\overline{BC}$ and $\overline{CD}$ are equal. If $m\angle ABC=84^\circ$ , which choice gives $m\angle BDC$ ?

For linked isosceles-triangle problems, assign a variable to the requested angle first. Then use each pair of equal sides to create equal angles, paying close attention to whether collinear points make two angles equal or supplementary. Finally, express the three angles of one triangle in terms of the variable and apply the triangle angle sum.

Hints

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Name the target angle

Let $m\angle BDC=x$ . Use $BC=CD$ to identify another angle that also measures $x$ .

Use the straight line

Since $A$ , $C$ , and $D$ are collinear, the angle at $C$ in triangle $ABC$ is supplementary to $\angle BCD$ .

Use the second pair of equal lengths

Rays $DA$ and $DC$ point in the same direction. Use this fact with $AB=BD$ to determine the angle at $A$ in triangle $ABC$ .

Desmos Guide

Set up the angle equation

An angle chase is faster than Desmos for this problem. After expressing the three angles of triangle $ABC$ as $x$ , $84$ , and $2x$ , the equation is $3x+84=180$ .

Graph both sides to verify

Enter $y=3x+84$ and $y=180$ . Select their intersection. Its $x$ -coordinate gives the measure of $\angle BDC$ .

Step-by-step Explanation

Use the isosceles triangle $BCD$

Let $m\angle BDC=x$ . Since $BC=CD$ , triangle $BCD$ is isosceles, so $m\angle CBD=x$ .

Therefore,

m\angle BCD=180^\circ-2x.

Relate the angles at $C$ and $A$

Because $A$ , $C$ , and $D$ are collinear, $\angle ACB$ and $\angle BCD$ are supplementary. Thus,

m\angle ACB=180^\circ-(180^\circ-2x)=2x.

Also, rays $DA$ and $DC$ are the same ray, so $m\angle ADB=m\angle CDB=x$ . Since $AB=BD$ , triangle $ABD$ is isosceles, giving $m\angle BAD=x$ .

Apply the triangle angle sum

The angles of triangle $ABC$ are $x$ , $84^\circ$ , and $2x$ . Therefore,

\begin{aligned} x+84+2x&=180 \\ 3x&=96 \\ x&=32. \end{aligned}

So $m\angle BDC=32^\circ$ .

Strategy

Hints

Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation