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Using Congruence in Proofs: Practice

12 multiple-choice questions, progressively harder.

Level 2 · Intermediate 0 / 12 answered

Question 1 of 12

1

A proof gives

$\triangle KLM\cong\triangle RST.$

If
$\begin{aligned}KL&=2x+7 \\ RS&=25\end{aligned}$
find $x$ .

Answer choices for question 1 A $\frac{25}{2}$ B $18$ C $9$ D $16$
2

The ordered congruence is:

$\triangle ABC\cong\triangle EFD$

The given angle measure is:

$m\angle F=68^\circ$

Find $m\angle B$ .

Answer choices for question 2 A $68^\circ$ B $34^\circ$ C $112^\circ$ D It cannot be determined
3

A four-sided figure $JKLM$ , with every corner pointing outward, has
$\begin{aligned}JK&=JM \\ KL&=ML\end{aligned}.$
Which auxiliary segment creates a pair of triangles that can be matched immediately by SSS?

Answer choices for question 3 A $KM$ B $JL$ C An arbitrary segment through K D No segment can help
4

After proving
$\begin{aligned}\triangle JKL&\cong\triangle JML\end{aligned}$
which equality is a valid CPCTC conclusion?

Answer choices for question 4 A $\angle JKL\cong\angle JLM$ B $JK=KL$ C $JL=KM$ D $\angle KJL\cong\angle MJL$
5

Points $A,B,D$ lie on one ray in that order, and $A,C,E$ lie on a second ray in that order. The two rays form an opening strictly between $0^\circ$ and $180^\circ$ . Given
$\begin{aligned}AB&=AC \\ AD&=AE\end{aligned}$
which ordered triangle pair is matched by SAS using the shared opening at $A$ ?

Answer choices for question 5 A $\triangle ABD\cong\triangle ACE$ B $\triangle ABE\cong\triangle ACD$ C $\triangle ABE\cong\triangle ADC$ D $\triangle ABC\cong\triangle ADE$
6

In the angle-copy construction,
$\begin{aligned}OA&=OB \\ OB&=PD \\ PD&=PE \\ PE&=11\text{ cm} \\ AB&=DE \\ DE&=13\text{ cm}\end{aligned}.$
Which pair of triangles is matched in the order needed to compare the angles at $O$ and $P$ ?

Answer choices for question 6 A $\triangle OAB\cong\triangle DPE$ B $\triangle OAB\cong\triangle DEP$ C $\triangle OAB\cong\triangle PDE$ D $\triangle OAB\cong\triangle EDP$
7

Triangles $ABC$ and $DBC$ share side $BC$ and have matching angles at $B$ and $C$ . If
$\begin{aligned}BD&=37\text{ m}\end{aligned}$
what is $AB$ in the exact model?

Answer choices for question 7 A $18.5$ m B $74$ m C It equals BC instead D $37$ m
8

A proof establishes
$\begin{aligned}\triangle PQR&\cong\triangle XYZ\end{aligned}.$
The corresponding lengths are
$\begin{aligned}PR&=2t+1 \\ XZ&=5t-20\end{aligned}.$
Find $t$ .

Answer choices for question 8 A $7$ B $\frac{19}{3}$ C $\frac{1}{7}$ D $21$
9

Two triangle drawings overlap. What must a separated redraw preserve for use in the proof?

Answer choices for question 9 A Any equal angles that the redraw now appears to show B The exact page positions of all vertices C An extra symmetry that makes the drawing easier D The vertex names, given marks, and intended correspondence
10

Lines $AC$ and $BD$ cross at $O$ , with
$\begin{aligned}OA&=OC \\ OB&=OD\end{aligned}.$
Which fact supplies the included angle pair for SAS in $\triangle AOB$ and $\triangle COD$ ?

Answer choices for question 10 A All angles around O are equal B Vertical angles are congruent C The two lines must be perpendicular D CPCTC applies before the triangles match
11

A two-stage congruence argument has established one pair of triangles and then, by CPCTC, a side pair needed for a second comparison. What must happen before CPCTC can be used on the second pair?

Answer choices for question 11 A Establish that second ordered triangle congruence from its own justified data B Assume every triangle in the figure is congruent C Use the first pair’s vertex order for every later triangle D Measure the diagram with a ruler
12

In an angle copy, which equality would be an invalid reason for SAS if it is the very result being proved?

Answer choices for question 12 A $OA=PD$ B $OB=PE$ C $\angle AOB\cong\angle DPE$ D $AB=DE$

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