Triangle Congruence Criteria - Math 1

1. In right triangles $\triangle DEF$ and $\triangle GHI$, both have right angles at $E$ and $H$ respectively. Given that $DE = GH$ and $EF = HI$, a student concludes the triangles are congruent by SAS. Is this conclusion correct?
   Answer: Yes, because we have two sides and the included right angle, which satisfies SAS The right angle at $E$ is indeed between sides $DE$ and $EF$, and the right angle at $H$ is between sides $GH$ and $HI$. Since $DE = GH$, $EF = HI$, and $\angle E = \angle H = 90°$, this satisfies SAS with the right angle as the included angle. Choice C mentions HL, but HL would require knowing which sides are the hypotenuses, and we'd need the hypotenuse and one leg, not two legs. Choice B incorrectly states that the right angle isn't between the given sides.
2. Triangle $ABC$ is isosceles with $AB = AC$. Point $D$ is the midpoint of $\overline{BC}$. To prove $\triangle ABD \cong \triangle ACD$ using SSS, which of the following must be established?
   Answer: $AB = AC$ (given), $BD = CD$ (since D is midpoint), and $AD = AD$ (reflexive property) SSS (Side-Side-Side) requires three pairs of equal corresponding sides. We have: (1) $AB = AC$ (given that triangle is isosceles), (2) $BD = CD$ (since $D$ is the midpoint of $\overline{BC}$), and (3) $AD = AD$ (reflexive property - the same side in both triangles). Choices A, C, and D incorrectly include angles, but SSS only involves sides, not angles.
3. Triangle $PQR$ has a right angle at $Q$. Triangle $STU$ has a right angle at $T$. If $PR = SU$ and $PQ = ST$, what additional information is needed to prove $\triangle PQR \cong \triangle STU$ using the Hypotenuse-Leg (HL) theorem?
   Answer: We need to verify that $PR$ and $SU$ are indeed the hypotenuses of their respective triangles For HL theorem to apply, we need the hypotenuse and one leg of each right triangle to be equal. We're told $PR = SU$ and $PQ = ST$, but we need to confirm which sides are the hypotenuses. In a right triangle, the hypotenuse is the side opposite the right angle. Since $\angle Q$ is the right angle in $\triangle PQR$, the hypotenuse is $PR$. Since $\angle T$ is the right angle in $\triangle STU$, the hypotenuse is $SU$. We need to verify this setup before applying HL.
4. Two triangles have all three pairs of corresponding angles equal. A student concludes that the triangles must be congruent. Which statement best evaluates this conclusion?
   Answer: The conclusion is incorrect; AAA only guarantees similarity, not congruence, unless additional side information is provided AAA (three pairs of equal corresponding angles) guarantees that triangles are similar, meaning they have the same shape but not necessarily the same size. For congruence, triangles must be both similar and have equal corresponding sides. You can have two triangles with all the same angles but different side lengths - for example, a 3-4-5 right triangle and a 6-8-10 right triangle have the same angles but different sizes. Congruence requires at least one pair of corresponding sides to be equal in addition to the angle relationships.
5. Two triangles have the following measurements: Triangle 1 has angles of $60°$, $70°$, and $50°$, with the side opposite the $60°$ angle measuring $8$ units. Triangle 2 has angles of $50°$, $60°$, and $70°$, with the side opposite the $50°$ angle measuring $8$ units. What can be concluded about these triangles?
   Answer: The triangles are not congruent because the given sides are not corresponding sides in the angle arrangement. While both triangles have the same three angles (50°, 60°, 70°), the sides given are not corresponding. In Triangle 1, the side of length 8 is opposite the 60° angle. In Triangle 2, the side of length 8 is opposite the 50° angle. Since angles of different measures have different opposite side lengths in any triangle, these sides are not equal to each other, so the triangles are not congruent. Choice D is wrong because AAA doesn't prove congruence.
6. Triangle $ABC$ is isosceles with $AB = AC$. Point $D$ is on side $\overline{BC}$ such that $\overline{AD} \perp \overline{BC}$. A student wants to prove $\triangle ABD \cong \triangle ACD$. Which reasoning contains an error?
   Answer: By SAS: $AB = AC$ (given), $\angle BAD = \angle CAD$ (base angles), and $AD = AD$ (reflexive property). Choice A contains an error in terminology. While ∠BAD = ∠CAD is true (the altitude from the vertex of an isosceles triangle bisects the vertex angle), these are not the 'base angles' of the isosceles triangle. The base angles are ∠ABD and ∠ACD. However, the SAS reasoning itself is actually valid since ∠BAD is between sides AB and AD, and ∠CAD is between sides AC and AD. The error is in the incorrect labeling, not the geometric reasoning.
7. Triangle $PQR$ is equilateral with side length $8$. Point $S$ is the centroid of the triangle. If we consider triangles $\triangle PSQ$, $\triangle QSR$, and $\triangle RSP$, which congruence criterion best establishes that all three triangles are congruent to each other?
   Answer: SAS, because each triangle has two sides of length 8 with a 120° angle between them at the centroid. The centroid divides the triangle into three smaller triangles. Each of these triangles has two sides that go from the centroid S to adjacent vertices of the equilateral triangle (like PS and QS), plus one side of the original triangle (like PQ). The distance from centroid to each vertex is the same, and the angle at the centroid in each small triangle is 360°/3 = 120°. So each triangle has two equal sides (centroid to vertex) with the included 120° angle. Choice A is wrong because the sides aren't all length 8. Choice C is wrong about which sides are shared. Choice D is wrong about the angle measures.
8. In right triangles $\triangle XYZ$ and $\triangle PQR$, $\angle Y$ and $\angle Q$ are the right angles. If $XZ = PR$ (the hypotenuses) and $YZ = QR$ (one pair of legs), which congruence criterion proves the triangles congruent?
   Answer: HL (Hypotenuse-Leg), because we have the hypotenuse and one leg congruent in both right triangles. HL (Hypotenuse-Leg) is the specific congruence criterion for right triangles when the hypotenuse and one leg are congruent. Choice A is incorrect because the right angles are not included between XZ and YZ. Choice C is incorrect because we don't know the third sides are equal yet (that would be the conclusion, not the reason). Choice D is incorrect because we only know one pair of angles (the right angles) are congruent.
9. Consider triangles $\triangle JKL$ and $\triangle MNP$ where $\angle J = \angle M = 40°$, $\angle K = \angle N = 85°$, and $JL = MN = 12$. Which statement about proving congruence is correct?
   Answer: The triangles cannot be proven congruent because the side JL does not correspond to side MN in the given angle configuration. In △JKL, the angles are ∠J=40°, ∠K=85°, so ∠L=55°. In △MNP, ∠M=40°, ∠N=85°, so ∠P=55°. For the triangles to be congruent with correspondence J↔M, K↔N, L↔P, we need JL to correspond to MP, not MN. The side JL connects vertices J and L, while MN connects M and N. These are not corresponding sides in this angle arrangement. We would need JL = MP to use AAS or ASA.
10. Given that $\triangle RST \cong \triangle UVW$ with the correspondence $R \leftrightarrow U$, $S \leftrightarrow V$, $T \leftrightarrow W$, which set of measurements would be sufficient to prove this congruence using SAS?
    Answer: $RS = UV$, $ST = VW$, and $\angle S = \angle V$, since we have two sides and the included angle. For SAS, we need two sides and the included angle. In choice A: RS corresponds to UV, ST corresponds to VW, and ∠S is the included angle between sides RS and ST, while ∠V is the included angle between UV and VW. Choice B has ∠R between RS and RT, but we're not given ST = VW. Choice C has ∠T, which is not included between RS and ST. Choice D has ∠S, but it's not included between ST and RT.
11. Given two triangles where $\triangle ABC$ has sides $AB = 7$, $BC = 10$, $AC = 12$ and $\triangle XYZ$ has sides $XY = 10$, $YZ = 12$, $XZ = 7$, a student writes $\triangle ABC \cong \triangle XYZ$ and justifies this with SSS. What error, if any, has the student made?
    Answer: The student should have written $\triangle ABC \cong \triangle YZX$ to show the correct correspondence of equal sides. The triangles do have the same side lengths (7, 10, 12), so they are congruent by SSS. However, the correspondence △ABC ≅ △XYZ implies AB=XY, BC=YZ, AC=XZ. But AB=7 while XY=10, so this correspondence is wrong. The correct correspondence should be AB=XZ=7, BC=XY=10, AC=YZ=12, giving △ABC ≅ △ZXY or △ABC ≅ △YZX. Choice A ignores the correspondence error. Choice C is wrong because SSS works regardless of listing order. Choice D is wrong because SSS doesn't require angle verification.
12. Triangle $PQR$ has vertices $P(2, 3)$, $Q(6, 3)$, and $R(4, 7)$. Triangle $STU$ has vertices $S(-1, 1)$, $T(-1, 5)$, and $U(-5, 3)$. Which congruence criterion can be used to prove these triangles are congruent?
    Answer: SSS, because all three pairs of corresponding sides have equal lengths after calculating distances. We need to calculate the side lengths. For triangle PQR: PQ = √[(6-2)² + (3-3)²] = 4, QR = √[(4-6)² + (7-3)²] = √20 = 2√5, PR = √[(4-2)² + (7-3)²] = √20 = 2√5. For triangle STU: ST = √[(-1-(-1))² + (5-1)²] = 4, TU = √[(-5-(-1))² + (3-5)²] = √20 = 2√5, SU = √[(-5-(-1))² + (3-1)²] = √20 = 2√5. Since all corresponding sides are equal (4, 2√5, 2√5), SSS proves congruence.
13. In triangles $\triangle FGH$ and $\triangle JKL$, we know that $\angle F = \angle J$, $\angle G = \angle K$, and $FH = JL$. A student claims the triangles are congruent by ASA. What is the issue with this claim?
    Answer: The triangles are congruent, but by AAS since the given side FH is not included between the two given angles F and G. For ASA, the side must be between (included between) the two given angles. Side FH connects vertices F and H, so it's between angles F and H, not between angles F and G. Since we have angles F and G, with side FH not between them, this is actually AAS (Angle-Angle-Side) where we have two angles and a non-included side. The triangles are still congruent, just by AAS rather than ASA. Choice A incorrectly states FH is opposite angle G. Choice B is wrong about correspondence. Choice C incorrectly suggests AAS and ASA are mutually exclusive.

Start this quiz