Math 1 Quiz: Applying Congruence Reasoning

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Applying Congruence ReasoningQuestion 1 of 7

In the coordinate plane, $\triangle PQR$ has vertices $P(2, 5)$ , $Q(8, 1)$ , and $R(4, 9)$ . Triangle $\triangle STU$ is obtained by reflecting $\triangle PQR$ across the line $y = x$ and then translating it 3 units down. If $\triangle PQR \cong \triangle STU$ , what are the coordinates of vertex $S$ ?

(5, -1)

(2, 5)

(5, 2)

(-1, 5)

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Math 1 Quiz

Math 1 Quiz: Applying Congruence Reasoning

Practice Applying Congruence Reasoning in Math 1 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Applying Congruence Reasoning, giving you a quick way to practice the rules, question types, and explanations that matter most for Math 1.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

$(5, -1)$ (correct answer)
$(2, 5)$
$(5, 2)$
$(-1, 5)$

Explanation: First, reflect

P(2, 5)

across

y = x

to get

(5, 2)

. Then translate 3 units down:

(5, 2-3) = (5, -1)

. Since reflections and translations preserve congruence,

S

corresponds to

P

under these transformations. Choice B gives the original coordinates. Choice C gives only the reflection. Choice D incorrectly applies the transformations.

Question 2

A bridge designer needs to create two identical triangular trusses. The first truss $\triangle ABC$ has been constructed with $\angle A = 65°$ , $\angle B = 45°$ , and side $AB = 24$ feet. For the second truss $\triangle DEF$ to be congruent to the first, which of the following sets of measurements would be sufficient?

$\angle D = 65°$ , $\angle E = 45°$ , and $DE = 24$ feet
$\angle D = 65°$ , $\angle F = 70°$ , and $DE = 24$ feet (correct answer)
$DE = 24$ feet, $EF = AC$ , and $DF = BC$
$\angle E = 45°$ , $DE = 24$ feet, and $EF = AC$

Explanation: In

\triangle ABC

\angle C = 180° - 65° - 45° = 70°

. Choice B gives

\angle D = 65°

(corresponding to

\angle A

\angle F = 70°

(corresponding to

\angle C

), and

DE = 24

feet (corresponding to

AB

). This satisfies AAS congruence (two angles and a non-included side). Choice A gives angle-angle-side but not in the correct AAS configuration. Choice C requires knowing the unknown lengths

AC

and

BC

. Choice D doesn't provide enough information for SAS congruence.

Question 3

Two right triangles, $\triangle PQR$ and $\triangle STU$ , have the following properties: $\angle P = \angle S = 90°$ , $PQ = ST$ , and $\angle Q = \angle T$ . A student claims these triangles are congruent by AAS. Which statement best evaluates this claim?

Correct, because two angles and a non-included side are equal in both triangles (correct answer)
Incorrect, because AAS requires two angles and the included side to be equal
Incorrect, because the given information actually demonstrates ASA congruence instead
Correct, but the triangles are also congruent by SAS using the right angles

Explanation: The student is correct. We have

\angle P = \angle S = 90°

\angle Q = \angle T

, and

PQ = ST

. Since

PQ

is not between the two given angles in triangle

PQR

, this is AAS congruence. Choice B confuses AAS with ASA. Choice C is wrong because we don't have ASA (the side isn't between the angles). Choice D is incorrect because we don't have two sides and an included angle.

Question 4

Two right triangles, $\triangle PQR$ and $\triangle STU$ , are congruent by the HL (Hypotenuse-Leg) theorem. In $\triangle PQR$ , the right angle is at $Q$ , $PR = 13$ , and $PQ = 5$ . In $\triangle STU$ , the right angle is at $T$ , and $SU = 13$ . If $\angle P = 22.6°$ , what is the measure of $\angle U$ ?

$22.6°$
$67.4°$ (correct answer)
$90°$
Cannot be determined without additional information

Explanation: In

\triangle PQR

, since

\angle Q = 90°

and

\angle P = 22.6°

, we have

\angle R = 180° - 90° - 22.6° = 67.4°

. By HL congruence, the triangles are congruent, but we need to determine the correct correspondence. Since both hypotenuses equal 13 (

PR = SU = 13

) and the right angles are at

Q

and

T

, the correspondence must be such that corresponding angles are equal. Since

\angle P

is opposite the shorter leg in

\triangle PQR

, the corresponding angle in

\triangle STU

would be

\angle U = 67.4°

Question 5

In parallelogram $WXYZ$ , diagonal $\overline{WY}$ creates two triangles. Using properties of parallelograms and congruence reasoning, which statement about $\triangle WXY$ and $\triangle WZY$ requires the most careful justification?

The triangles are congruent by SSS because $WX = YZ$ , $XY = WZ$ , and $WY = WY$
The triangles are congruent by SAS because $WX = YZ$ , $WZ = XY$ , and $\angle XWY = \angle ZYW$ (correct answer)
The triangles have equal areas but are not necessarily congruent without additional information
The triangles are congruent by ASA because alternate interior angles are equal when parallel lines are cut by a transversal

Explanation: Option B requires the most careful justification because it incorrectly applies SAS congruence. While

WX = YZ

and

WZ = XY

(opposite sides of parallelogram) and

\angle XWY = \angle ZYW

(alternate interior angles), these don't form a valid SAS correspondence. For SAS, the angle must be included between the two given sides. Here,

\angle XWY

is not between sides

WX

and

WZ

. The triangles are congruent, but this particular reasoning is flawed.

Question 6

A carpenter is cutting two triangular pieces of wood. The first triangle has sides of length $a$ , $b$ , and $c$ . The second triangle has sides of length $b$ , $c$ , and $a$ (in that order). Using congruence reasoning, which statement is most accurate?

The triangles are congruent by SSS since they have the same three side lengths (correct answer)
The triangles are congruent by SAS since two sides and the included angle are equal
The triangles are similar but not necessarily congruent due to different side arrangements
The triangles cannot be determined to be congruent without angle measurements

Explanation: Both triangles have sides of lengths

a

b

, and

c

. The order in which the sides are listed doesn't matter for congruence - what matters is that both triangles have the same set of side lengths. By the SSS (Side-Side-Side) congruence theorem, if three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.

Question 7

Two overlapping triangles $\triangle PQR$ and $\triangle PST$ share vertex $P$ and side $\overline{PQ}$ . Given that $PQ = PS$ , $\angle QPR = \angle SPT$ , and $PR = PT$ , a student concludes the triangles are congruent by SAS. However, upon closer inspection of the angle relationship, what is the correct congruence theorem that applies?

SAS congruence is correct since $PQ = PS$ , $\angle QPR = \angle SPT$ , and $PR = PT$
SSS congruence applies because we can show $QR = ST$ using the given information
The triangles are congruent by SAS, but using $\angle QPS = \angle RPS$ as the included angle instead
No congruence theorem applies because $\angle QPR$ and $\angle SPT$ are not included angles (correct answer)

Explanation: For SAS congruence, we need two sides and the included angle between them. Here we have

PQ = PS

and

PR = PT

, but

\angle QPR

and

\angle SPT

are not the included angles between these pairs of sides. The included angle for sides

PQ

and

PR

would be

\angle QPR

, and for sides

PS

and

PT

would be

\angle SPT

, but these are not equal to each other in the way needed for SAS. We would need

\angle QPS = \angle RPT

or find another approach.