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Using Congruence in Proofs: Practice

12 multiple-choice questions, progressively harder.

Level 3 · Challenge 0 / 12 answered

Question 1 of 12

1

Lines $AC$ and $BD$ cross at $O$ . Given
$\begin{aligned}OA&=OC \\ OC&=14 \\ OB&=OD \\ OD&=19\end{aligned}$
which conclusion follows by SAS then CPCTC?

Answer choices for question 1 A $AB=CD$ B $AB=AD$ C $OA=OB$ D The crossing is perpendicular
2

A four-sided figure $PQRS$ , with every corner pointing outward, has
$\begin{aligned}PQ&=PS \\ RQ&=RS\end{aligned}.$
After drawing $PR$ , what proves that it bisects the interior angle at $P$ ?

Answer choices for question 2 A CPCTC establishes triangle congruence directly B The diagonal makes both angles right C SSS for triangles PQR and PSR, followed by CPCTC at P D SSS proves the four-sided figure is a square
3

In $\triangle ABC$ and $\triangle DEF$ ,
$\begin{aligned}AB&=DE \\ \angle A&\cong\angle D \\ \angle C&\cong\angle F\end{aligned}.$
Which criterion establishes congruence before the following conclusion?
$\begin{aligned}BC&=EF\end{aligned}.$

Answer choices for question 3 A SAS B SSS C HL D AAS
4

A proof establishes
$\begin{aligned}\triangle LMN&\cong\triangle UVW\end{aligned}.$
If
$\begin{aligned}m\angle N&=4x+3\text{ degrees} \\ m\angle W&=7x-18\text{ degrees}\end{aligned}$
find the common angle measure.

Answer choices for question 4 A $7^\circ$ B $31^\circ$ C $28^\circ$ D $49^\circ$
5

In an indirect-distance model, the accessible triangle reproduces only one sighting angle and the shared baseline. Why can the target length not yet be transferred by ASA?

Answer choices for question 5 A ASA requires all three side lengths B A shared side can never be used C CPCTC supplies the missing angle before congruence D A second matching angle is missing
6

Two right triangles have right angles at $B$ and $E$ , with
$\begin{aligned}AC&=DF \\ AB&=DE\end{aligned}.$
After matching them by HL, which side pair may be concluded by CPCTC?

Answer choices for question 6 A $AB=EF$ B $BC=EF$ C $AC=DE$ D $BC=DF$
7

A proof has established
$\begin{aligned}\triangle ABC&\cong\triangle DEF\end{aligned}.$
Which equality compares the angles opposite corresponding sides $AB$ and $DE$ ?

Answer choices for question 7 A $\angle ACB\cong\angle DFE$ B $\angle ABC\cong\angle DFE$ C $\angle BAC\cong\angle DFE$ D $\angle ACB\cong\angle DEF$
8

A compass procedure preserves the two radii of a source triangle but changes the transferred chord length. Which conclusion is justified?

Answer choices for question 8 A The copied angle is unchanged for every chord length B The triangles remain congruent by CPCTC alone C The SSS angle-copy argument no longer applies to those data D The chord does not affect the angle-copy proof
9

In a figure,
$\begin{aligned}\triangle ABE&\cong\triangle ACD\end{aligned}$
has been proved. If
$\begin{aligned}BE&=3z+8 \\ CD&=5z-10\end{aligned}$
find the common side length.

Answer choices for question 9 A $9$ B $27$ C $35$ D $45$
10

A diagonal, a segment joining opposite corners, is drawn in a four-sided figure with every corner pointing outward. Which claim is available solely because the diagonal was drawn?

Answer choices for question 10 A It bisects both vertex angles B It bisects the other diagonal C It is perpendicular to every side D It is a side of each of the two triangles it creates
11

Suppose a first triangle congruence proves
$\begin{aligned}TU&=VW\end{aligned}.$
A second pair has
$\begin{aligned}RT&=SV \\ RU&=SW\end{aligned}.$
What criterion now establishes the following congruence?
$\begin{aligned}\triangle RTU&\cong\triangle SVW\end{aligned}.$

Answer choices for question 11 A AAA B SSS C SSA D CPCTC without a criterion
12

Which statement best explains why a copied angle can be justified without a protractor measurement?

Answer choices for question 12 A The construction transfers three lengths, SSS matches the triangles, and CPCTC transfers the angle B Compass arcs prove all angles are equal C Equal radii alone force the source and target angles to be equal D The copied angle is assumed equal when drawing begins

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