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SSS and SAS Similarity: Practice

12 multiple-choice questions, progressively harder.

Level 2 · Intermediate 0 / 12 answered

Question 1 of 12

1

Two triangles have matching side lengths $9, 10, 14$ and $13.5, 15, 21$ . Which conclusion follows?

Answer choices for question 1 A They are congruent B Two ratios agree but the third fails C An angle measurement is still required D They are similar by SSS with factor 1.5
2

Triangle $ABC$ has
$\begin{aligned}AB&=8\\BC&=10\\CA&=12.\end{aligned}$
Triangle $DEF$ has
$\begin{aligned}DE&=10\\EF&=12\\FD&=8.\end{aligned}$
Which ordered similarity is justified by SSS?

Answer choices for question 2 A $\triangle ABC\sim\triangle DEF$ B $\triangle ABC\sim\triangle FDE$ C $\triangle ABC\sim\triangle EFD$ D $\triangle ABC\sim\triangle DFE$
3

Sides
$\begin{aligned}AB&=6\\AC&=10\end{aligned}$
meet at a $53^\circ$ angle. Sides
$\begin{aligned}DE&=9\\DF&=15\end{aligned}$
meet at the same angle. If $BC=t$ , what is $EF$ ?

Answer choices for question 3 A $\frac23t$ B $t+3$ C $1.5t$ D It cannot be related to t
4

The ratios $DE/AB$ and $EF/BC$ agree. Which added angle equality would establish SAS similarity for the order $ABC$ to $DEF$ ?

Answer choices for question 4 A $\angle ABC\cong\angle DEF$ B $\angle BAC\cong\angle EDF$ C $\angle ACB\cong\angle DFE$ D $\angle BAC\cong\angle DFE$
5

For the correspondence $ABC$ to $DEF$ , the side lengths are
$\begin{aligned}AB&=6\\BC&=8\\CA&=11\\DE&=9\\EF&=12\\FD&=x.\end{aligned}$
What value of $x$ makes SSS similarity hold?

Answer choices for question 5 A $14$ B $16.5$ C $15$ D $22$
6

SAS establishes similarity with factor $\frac 85$ from triangle $KLM$ to triangle $PQR$ . What is the factor from $PQR$ back to $KLM$ ?

Answer choices for question 6 A $\frac85$ B $\frac35$ C $\frac83$ D $\frac58$
7

Both within-triangle ratios $AB/BC$ and $DE/EF$ equal $3/4$ , and $\angle ABC\cong\angle DEF$ . Which conclusion is correct?

Answer choices for question 7 A SAS gives similarity, but these within-triangle ratios alone do not specify the between-triangle factor B The scale from ABC to DEF is necessarily three quarters C The scale from ABC to DEF is necessarily four thirds D These equalities contradict SAS
8

Triangles satisfy $\angle A\cong\angle D$ , with
$\begin{aligned}AB&=x+4\\AC&=10\\DE&=18\\DF&=15.\end{aligned}$
What value of $x$ allows SAS similarity in the stated order?

Answer choices for question 8 A $6$ B $11$ C $8$ D $12$
9

A SAS similarity has side factor $0.8$ and an included angle measuring $44^\circ$ in the original triangle. What is the corresponding image angle?

Answer choices for question 9 A $35.2^\circ$ B $55^\circ$ C $44^\circ$ D $88^\circ$
10

A side-ratio test fails for one proposed vertex matching. What can safely be concluded before other matchings have been considered?

Answer choices for question 10 A That proposed matching fails; another matching may still work B The triangles cannot be similar under any matching C All their corresponding angles must differ D SSS similarity is false as a theorem
11

In triangle $ABC$ , $D$ lies between $A$ and $B$ and $E$ lies between $A$ and $C$ . If
$\begin{aligned}AD&=6\\AB&=14\\AE&=9\\AC&=21\end{aligned}$
, which criterion establishes $\triangle ADE\sim\triangle ABC$ without a parallelism assumption?

Answer choices for question 11 A SSS, using an unmeasured third side B AA, with only the angle at A C No criterion can apply D SAS, using the shared angle at A
12

Points $A,O,C$ and $B,O,D$ are collinear in those orders. If
$\begin{aligned}AO&=5\\CO&=12\\BO&=7\\DO&=16.8\end{aligned}$
, why is $\triangle AOB\sim\triangle COD$ justified without assuming $AB\parallel CD$ ?

Answer choices for question 12 A SSS, because the outer sides are shared B SAS, using proportional sides around equal vertical angles at O C AA, because crossing lines give two distinct angle pairs D The outer sides must be assumed equal

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