Super-Hard SAT Geometry Question 16 | Free SAT Prep

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Question 16·25 Super-Hard SAT Geometry and Trigonometry Questions·Geometry and Trigonometry

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For triangles $ABC$ and $DEF$ , angles $B$ and $E$ each measure $48^\circ$ , the length of $\overline{AB}$ is $14$ , and the length of $\overline{DE}$ is $21$ . Which of the following additional pieces of information are sufficient to prove that triangle $ABC$ is similar to triangle $DEF$ ?

I. The length of $\overline{BC}$ is $16$ , and the length of $\overline{EF}$ is $24$ .

II. The length of $\overline{AC}$ is $12$ , and the length of $\overline{DF}$ is $18$ .

III. An exterior angle at $A$ has the same measure as an exterior angle at $D$ .

For triangle-similarity sufficiency questions, first match the vertices using the given equal angles. Then check whether two proportional side pairs surround those equal angles; that is SAS. Two equal angles establish AA, including when equal exterior angles imply equal supplementary interior angles. Be cautious when a side is opposite the known angle: proportional sides plus a nonincluded equal angle is SSA and does not guarantee similarity.

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Match sides to the given angles

For an SAS argument, both side pairs must meet at the already-known equal angles $B$ and $E$ .

Check the location of each side

Notice whether each stated side is adjacent to the given angle or opposite it. Two proportional sides are not automatically enough for similarity.

Relate exterior and interior angles

At a vertex of a triangle, an exterior angle and the interior angle form a straight angle. Consider what equal exterior angles imply about their interior angles.

Desmos Guide

Use geometry first

The fastest method is to identify SAS, SSA, and AA directly. Desmos is most useful here for verifying why statement II can be ambiguous.

Model the SSA information

Place $B$ at $(0,0)$ and $A$ at $(14,0)$ . Enter y=x tan(48°) {x>=0} to represent the ray from $B$ that forms a $48^\circ$ angle with $\overline{BA}$ .

Locate possible positions of C

Enter (x-14)^2+y^2=144, the circle centered at $A$ with radius $12$ . The ray and circle have two intersection points, showing that the SSA information in statement II can create two different triangle shapes rather than one required similar shape.

Step-by-step Explanation

Evaluate statement I using SAS similarity

The sides that form $\angle B$ are $\overline{AB}$ and $\overline{BC}$ , and the sides that form $\angle E$ are $\overline{DE}$ and $\overline{EF}$ .

\frac{AB}{DE}=\frac{14}{21}=\frac{2}{3} \qquad\text{and}\qquad \frac{BC}{EF}=\frac{16}{24}=\frac{2}{3}

These proportional side pairs include the equal angles $B$ and $E$ , so statement I is sufficient by SAS similarity.

Evaluate statement II carefully

Statement II gives $\frac{AB}{DE}=\frac{14}{21}=\frac{2}{3}$ and $\frac{AC}{DF}=\frac{12}{18}=\frac{2}{3}$ . However, $\overline{AC}$ and $\overline{DF}$ are opposite the given equal angles $B$ and $E$ ; they do not form those angles with $\overline{AB}$ and $\overline{DE}$ .

This is an SSA arrangement, which can produce triangles with different shapes. Therefore, statement II is not sufficient.

Use the exterior angles in statement III

An exterior angle at $A$ is supplementary to $\angle A$ , and an exterior angle at $D$ is supplementary to $\angle D$ . If the exterior angles have equal measure, then their supplementary interior angles also have equal measure. Thus, $\angle A=\angle D$ .

Together with $\angle B=\angle E$ , this gives two pairs of equal angles. Therefore, statement III is sufficient by AA similarity.

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Step-by-step Explanation

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Step-by-step Explanation