Applying Congruence Reasoning - Math 1

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?
Answer: The triangles are congruent by SSS since they have the same three side lengths 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.
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 $\angle U$ ?
Answer: $67.4°$ 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 $\angle U = 67.4°$ .
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 $S$ ?
Answer: $(5, -1)$ 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.
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?
Answer: No congruence theorem applies because $\angle QPR$ and $\angle SPT$ are not included angles 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.
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?
Answer: $\angle D = 65°$ , $\angle F = 70°$ , and $DE = 24$ feet 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.
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?
Answer: The triangles are congruent by SAS because $WX = YZ$ , $WZ = XY$ , and $\angle XWY = \angle ZYW$ 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.
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?
Answer: Correct, because two angles and a non-included side are equal in both triangles 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.

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