Unit 4 Congruent Triangles Homework 1 Answer Key

Unit 4 Congruent Triangles Homework 1: A Comprehensive Guide with Answers

This guide provides comprehensive solutions and explanations for Unit 4 Congruent Triangles Homework 1. Understanding congruent triangles is crucial in geometry, forming the basis for many advanced theorems and applications. This guide aims to not only provide the answers but also to deepen your understanding of the underlying concepts. We'll tackle various problems, explaining the reasoning behind each step and highlighting key theorems and postulates. Remember, understanding the why is as important as knowing the what.

Understanding Congruent Triangles

Before diving into the homework problems, let's refresh our understanding of congruent triangles. Two triangles are congruent if their corresponding sides and angles are equal. This means that if triangle ABC is congruent to triangle DEF (written as ΔABC ≅ ΔDEF), then:

• AB = DE
• BC = EF
• AC = DF
• ∠A = ∠D
• ∠B = ∠E
• ∠C = ∠F

Several postulates and theorems help us determine if two triangles are congruent without having to check all six congruences. The most common ones include:

• SSS (Side-Side-Side): If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
• SAS (Side-Angle-Side): If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
• ASA (Angle-Side-Angle): If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.
• AAS (Angle-Angle-Side): If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent.
• HL (Hypotenuse-Leg): This postulate applies only to right-angled triangles. If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.

Homework Problems and Solutions

Let's tackle some typical problems found in Unit 4 Congruent Triangles Homework 1. Remember, the specific problems in your homework might vary, but the principles applied will remain the same. We'll cover a range of difficulty levels.

Problem 1: Basic Congruence

Given: ΔABC and ΔDEF with AB = DE, BC = EF, AC = DF.

Question: Prove ΔABC ≅ ΔDEF.

Solution: This problem directly applies the SSS postulate. Since all three sides of ΔABC are congruent to the corresponding three sides of ΔDEF, we can conclude that ΔABC ≅ ΔDEF by SSS.

Problem 2: Using SAS

Given: ΔPQR and ΔXYZ with PQ = XY, ∠Q = ∠Y, QR = YZ.

Question: Prove ΔPQR ≅ ΔXYZ.

Solution: This problem uses the SAS postulate. We have two sides (PQ = XY and QR = YZ) and the included angle (∠Q = ∠Y) congruent. Therefore, ΔPQR ≅ ΔXYZ by SAS.

Problem 3: Applying ASA

Given: ΔLMN and ΔOPQ with ∠L = ∠O, LM = OP, ∠M = ∠P.

Question: Prove ΔLMN ≅ ΔOPQ.

Solution: This uses the ASA postulate. We have two angles (∠L = ∠O and ∠M = ∠P) and the included side (LM = OP) congruent. Thus, ΔLMN ≅ ΔOPQ by ASA.

Problem 4: A Challenge with AAS

Given: ΔRST and ΔUVW with ∠R = ∠U, ∠T = ∠W, ST = VW.

Question: Prove ΔRST ≅ ΔUVW.

Solution: This employs the AAS postulate. We're given two angles (∠R = ∠U and ∠T = ∠W) and a non-included side (ST = VW). Since the third angle in each triangle is determined by the sum of angles in a triangle being 180°, we implicitly know the third angles are also equal. Therefore, ΔRST ≅ ΔUVW by AAS.

Problem 5: Right-Angled Triangles and HL

Given: Right-angled triangles ΔABC and ΔDEF, where ∠B = ∠E = 90°, AC = DF (hypotenuse), and BC = EF (leg).

Question: Prove ΔABC ≅ ΔDEF.

Solution: This problem utilizes the HL postulate specific to right-angled triangles. The hypotenuse (AC = DF) and a leg (BC = EF) are congruent. Therefore, ΔABC ≅ ΔDEF by HL.

Problem 6: Multi-Step Proof

This problem involves a more complex scenario requiring multiple steps and combining different postulates. Let's consider a case where we have two triangles sharing a common side.

Given: In the diagram, AB = AC, and BD = CD. Prove that ΔABD ≅ ΔACD.

Solution: This problem requires a step-by-step approach:

1. AB = AC (Given): This is stated in the problem.
2. BD = CD (Given): Also stated in the problem.
3. AD = AD (Reflexive Property): A side is congruent to itself.
4. ΔABD ≅ ΔACD (SSS): Since all three sides of ΔABD are congruent to the corresponding sides of ΔACD, the triangles are congruent by SSS.

Problem 7: Application in Problem Solving

Let's apply our knowledge of congruent triangles to a real-world problem.

Given: You have two triangular pieces of wood, each with angles measuring 45°, 45°, and 90°. One piece has legs of length 5cm, and the other has legs of length 10cm. Are the triangles congruent?

Solution: While the angles are the same (45°, 45°, 90°), making them similar triangles, they are not congruent because the lengths of the legs are different. Congruent triangles require corresponding sides to be equal.

Further Practice and Advanced Concepts

This guide has covered fundamental concepts and problem types related to congruent triangles. To further solidify your understanding, engage in more practice problems. Explore problems involving:

• Isosceles triangles: Utilize the properties of isosceles triangles (two equal sides, two equal angles).
• Proofs involving multiple triangles: These problems often require proving congruence of one pair of triangles to facilitate the proof of congruence for another pair.
• Coordinate geometry and congruent triangles: Use coordinate geometry to prove triangle congruence.

Remember, mastering congruent triangles is crucial for progressing to more complex geometric concepts. Consistent practice and a deep understanding of the postulates and theorems will build your confidence and competence in tackling challenging problems. Remember to always clearly state which postulate or theorem you are using in your proofs. This enhances clarity and makes your reasoning easier to follow.