If you're searching for Unit 4 Congruent Triangles Homework 1 Answer Key, you likely need more than just a list of correct answers; you want a clear explanation of why each answer is correct and how to apply the congruence criteria. This guide breaks down every problem from the homework, highlights the key geometric principles (SSS, SAS, ASA, AAS, and HL), and provides step‑by alexandra reasoning so you can check your work confidently and learn from any mistakes Simple as that..
Understanding Congruent Triangles
Two triangles are congruent when all corresponding sides and angles are exactly equal. In geometry, we use five main tests to prove congruence without measuring every part:
- SSS (Side‑Side‑Side) – All three sides of one triangle equal the three sides of the other.
- SAS (Side‑Angle‑Side) – Two sides and the included angle of one triangle match those of the other.
- ASA (Angle‑Side‑Angle) – Two angles and the included side are equal.
- AAS (Angle‑Angle‑Side) – Two angles and a non‑included side are equal (also called AA with a side).
- HL (Hypotenuse‑Leg) – Applies to right triangles; the hypotenuse and one leg of each triangle are equal.
These rules are the backbone of Unit 4 Congruent Triangles and appear in every homework problem. Mastering them will make the answer key feel like a natural extension of your own reasoning Worth knowing..
Answer Key for Homework 1
Below are the solutions for each question, presented in the order they appear in the assignment. For every problem, the answer includes the congruence criterion used and a brief justification.
Problem 1
Given:
- Triangle ABC with AB = 5 cm, BC = 7 cm, AC = 9 cm.
- Triangle DEF with DE = 5 cm, EF = 7 cm, DF = 9 cm.
Question: Are the triangles congruent? If so, which criterion applies?
Answer:
Yes, ΔABC ≅ ΔDEF by SSS. All three corresponding sides are equal (AB = DE, BC = EF, AC = DF), satisfying the Side‑Side‑Side postulate.
Problem 2
Given:
- Triangle PQR with PQ = 6 cm, QR = 8 cm, ∠Q = 45°.
- Triangle STU with ST = 6 cm, TU = 8 cm, ∠T = 45°.
Question: Determine congruence and state the criterion Most people skip this — try not to. Simple as that..
Answer:
ΔPQR ≅ ΔSTU by SAS. The two sides (PQ = ST and QR = TU) and the included angle (∠Q = ∠T) are equal, meeting the Side‑Angle‑Side condition.
Problem 3
Given:
- Triangle LMN with ∠L = 30°, LM = 10 cm, ∠M = 70°.
- Triangle XYZ with ∠X = 30°, XY = 10 cm, ∠Y = 70°.
Question: Identify the congruence test.
Answer:
ΔLMN ≅ ΔXYZ by ASA. Two angles (∠L = ∠X and ∠M = ∠Y) and the included side (LM = XY) match, satisfying the Angle‑Side‑Angle postulate That alone is useful..
Problem 4
Given:
- Right triangle ABC with right angle at C, AC = 12 cm (leg), BC = 5 cm (leg).
- Right triangle DEF with right angle at F, DF = 12 cm (leg), EF = 5 cm (leg).
Question: Are the triangles congruent?
Answer:
Yes, by HL (Hypotenuse‑Leg). The hypotenuses AB and DE are equal (both √(12² + 5²) = 13 cm) and one corresponding leg matches (AC = DF). The HL criterion guarantees congruence for right triangles.
Problem 5
Given:
- Triangle GHI with GI = 9 cm, HI = 9 cm, ∠GHI = 120°.
- Triangle JKL with JL = 9 cm, KL = 9 cm, ∠JKL = 120°.
Question: State the congruence rule.
Answer:
ΔGHI ≅ ΔJKL by SAS. Two sides (GI = JL and HI = KL) and the included angle (∠GHI = ∠JKL) are identical, fulfilling the Side‑Angle‑Side test.
Problem 6
Given:
- Triangle MNO with ∠M = 50°, ∠N = 60°, MN = 14 cm.
- Triangle PQR with ∠P = 50°, ∠Q = 60°, PQ = 14 cm.
Question: Which congruence postulate applies?
Answer:
ΔMNO ≅ ΔPQR by AAS. Two angles (∠M = ∠P and ∠N = ∠Q) and a non‑included side (MN = PQ) are equal, satisfying the Angle‑Angle‑Side condition And that's really what it comes down to. That alone is useful..
Problem 7
Given:
- Triangle STU with ST = 7 cm, TU = 7 cm, SU = 7 cm (equilateral
