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Similarity and AA: Practice

12 multiple-choice questions, progressively harder.

Level 3 · Challenge 0 / 12 answered

Question 1 of 12

1

A proposed AA proof uses the angle sum to find equal third angles and immediately concludes that the sides are proportional. What step is missing?

Answer choices for question 1 A Add the three angle measures again B Assume the drawing is to scale C Scale a corresponding side to match, then justify congruence using the preserved angles D Declare every pair of triangles congruent
2

Triangle $ABC$ has angles $48^\circ$ , $67^\circ$ , $65^\circ$ at $A,B,C$ . Triangle $PQR$ has angles $65^\circ$ , $48^\circ$ , $67^\circ$ at $P,Q,R$ . Which ordered statement is valid?

Answer choices for question 2 A $\triangle ABC\sim\triangle PQR$ B $\triangle BCA\sim\triangle PQR$ C $\triangle ACB\sim\triangle PQR$ D $\triangle CAB\sim\triangle PQR$
3

In triangle $ABC$ , $D$ is between $A$ and $B$ , $E$ is between $A$ and $C$ , and $DE\parallel BC$ . If
$\begin{aligned}AD&=8\\DB&=12\\AE&=6\end{aligned}$
, find $AC$ .

Answer choices for question 3 A $9$ B $15$ C $10$ D $24$
4

Points $A,O,C$ and $B,O,D$ are collinear in order, with $AB\parallel CD$ . If
$\begin{aligned}AO&=12\\CO&=18\\DO&=10.5\end{aligned}$
, find $BO$ .

Answer choices for question 4 A $7$ B $15.75$ C $4.5$ D $9$
5

Triangle $ABC$ has angles $46^\circ$ , $67^\circ$ , $67^\circ$ at $A,B,C$ . Triangle $DEF$ has angles $67^\circ$ , $46^\circ$ , $67^\circ$ at $D,E,F$ . Which correspondence is forced in every valid similarity statement?

Answer choices for question 5 A $B$ must match $D$ B $C$ must match $D$ C $A$ must match $D$ D $A$ must match $E$
6

Two rectangles have all four corresponding angles equal. Their corresponding side lengths are 2 and 3 in the first rectangle, and 2 and 5 in the second. What follows?

Answer choices for question 6 A They are not similar under this matching because the side ratios differ B They must be similar by AA C They must be congruent because all angles match D Every quadrilateral obeys the triangle AA criterion
7

In an AA proof, a dilation has made the side between two preserved angles equal to the corresponding side of the target. Why is it valid to use ASA at this point?

Answer choices for question 7 A ASA similarity was assumed at the start B Equal angles alone already made the original triangles congruent C ASA is an earlier congruence result about two angles and their included side D The two triangles have necessarily been drawn in the same position
8

Two right triangles have their right angles at $B$ and $E$ . The first has $A=28^\circ$ ; the second has $D=62^\circ$ . Which ordered similarity statement follows?

Answer choices for question 8 A $\triangle ABC\sim\triangle DEF$ B $\triangle ABC\sim\triangle FED$ C $\triangle ABC\sim\triangle EFD$ D The angle information is insufficient
9

For nondegenerate triangles,
$\begin{aligned}\angle A\cong\angle D\\\angle B\cong\angle E.\end{aligned}$
After establishing AA, the lengths $AB$ and $DE$ are $2x+5$ and $5x-1$ , and the scale from $ABC$ to $DEF$ is $\frac 32$ . Find $x$ .

Answer choices for question 9 A $\frac{17}{4}$ B $\frac94$ C $\frac{13}{2}$ D $\frac{17}{2}$
10

Known corresponding angles satisfy
$\begin{aligned}m\angle A&=39^\circ\\m\angle D&=(x+19)^\circ\\m\angle B&=67^\circ\\m\angle E&=(2x+27)^\circ.\end{aligned}$
Which value makes both angle pairs equal for AA?

Answer choices for question 10 A $28$ B $40$ C $20$ D $10$
11

It is given that
$\begin{aligned}\triangle ABC\sim\triangle XYZ\\AB&=XY.\end{aligned}$
Which conclusion follows for the stated correspondence?

Answer choices for question 11 A Only AB and XY can be equal B Every corresponding side is equal, so the triangles are congruent C The scale factor may still be any positive number D The triangles must have the same vertex coordinates
12

Triangles $ABC$ and $DEF$ have equal angle pairs $A$ to $D$ and $B$ to $E$ . The dilation factor from $ABC$ to $DEF$ is $\frac 54$ . Which equation correctly gives the reverse mapping of side $EF$ ?

Answer choices for question 12 A $BC=\frac54\,EF$ B $AB=\frac45\,EF$ C $BC=EF-\frac14$ D $BC=\frac45\,EF$

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