Unit 5 Relationships In Triangles Homework 2 Answer Key

Unit 5: Relationships in Triangles - Homework 2 Answer Key: A Comprehensive Guide

This comprehensive guide provides detailed solutions and explanations for Homework 2 in Unit 5: Relationships in Triangles. We'll cover key concepts, theorems, and problem-solving strategies to help you master this unit. Understanding these relationships is crucial for success in higher-level geometry and related mathematical fields.

Understanding the Fundamentals: Key Theorems and Concepts

Before diving into the answers, let's refresh our understanding of the fundamental theorems and concepts related to triangles. A strong grasp of these is essential for solving the problems in Homework 2.

1. Triangle Inequality Theorem:

This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This helps determine if a given set of side lengths can actually form a triangle.

2. Pythagorean Theorem:

Applicable only to right-angled triangles, this theorem states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (legs). This is fundamental for calculating side lengths in right-angled triangles.

3. Similar Triangles:

Two triangles are similar if their corresponding angles are congruent (equal) and their corresponding sides are proportional. Similarity theorems, such as AA (Angle-Angle), SAS (Side-Angle-Side), and SSS (Side-Side-Side), help establish similarity.

4. Congruent Triangles:

Two triangles are congruent if their corresponding sides and angles are congruent. Congruence theorems, such as ASA (Angle-Side-Angle), SAS (Side-Angle-Side), SSS (Side-Side-Side), and AAS (Angle-Angle-Side), are used to prove congruence.

5. Special Right Triangles:

Understanding the properties of 30-60-90 and 45-45-90 triangles simplifies calculations significantly. These triangles have specific ratios between their sides.

Homework 2 Problems and Solutions: A Step-by-Step Approach

Now, let's delve into the problems and solutions for Homework 2. We'll assume a typical homework assignment structure, breaking down each problem with detailed explanations and reasoning. Remember, the specific problems in your assignment might vary slightly, but the underlying principles and techniques remain the same.

Problem 1: Determining Triangle Validity

Problem: Can a triangle have sides with lengths 5, 7, and 13?

Solution: Apply the Triangle Inequality Theorem.

• 5 + 7 > 13? (No, 12 < 13)
• 5 + 13 > 7? (Yes, 18 > 7)
• 7 + 13 > 5? (Yes, 20 > 5)

Since one inequality is false, a triangle with these side lengths is not possible.

Problem 2: Using the Pythagorean Theorem

Problem: A right-angled triangle has legs of length 6 and 8. Find the length of the hypotenuse.

Solution: Use the Pythagorean Theorem: a² + b² = c²

• a = 6, b = 8
• 6² + 8² = c²
• 36 + 64 = c²
• 100 = c²
• c = √100 = 10

The length of the hypotenuse is 10.

Problem 3: Similar Triangles

Problem: Two triangles, ΔABC and ΔDEF, are similar. If AB = 4, BC = 6, and DE = 8, find the length of EF.

Solution: Since the triangles are similar, their corresponding sides are proportional.

• AB/DE = BC/EF
• 4/8 = 6/EF
• EF = (6 * 8) / 4 = 12

The length of EF is 12.

Problem 4: Congruent Triangles

Problem: Prove that two triangles are congruent given that two sides and the included angle are congruent in both triangles.

Solution: This demonstrates the SAS (Side-Angle-Side) congruence postulate. 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. A formal proof would involve stating the given information, the congruence postulate, and the conclusion.

Problem 5: Special Right Triangles

Problem: Find the lengths of the other two sides of a 30-60-90 triangle if the hypotenuse is 10.

Solution: In a 30-60-90 triangle, the ratio of sides opposite the angles is 1:√3:2.

• Hypotenuse = 2x = 10 => x = 5
• Side opposite 30° = x = 5
• Side opposite 60° = x√3 = 5√3

The lengths are 5 and 5√3.

Problem 6: Applying Multiple Concepts

Problem: A triangle has sides of length 5, 12, and 13. Is it a right-angled triangle? If so, what is its area?

Solution: Check if the Pythagorean Theorem holds:

• 5² + 12² = 25 + 144 = 169
• 13² = 169

Since the theorem holds, it's a right-angled triangle. The area is:

• Area = (1/2) * base * height = (1/2) * 5 * 12 = 30

The area is 30 square units.

Problem 7: Word Problem involving Triangles

Problem: A ladder leans against a wall. The base of the ladder is 6 feet from the wall, and the top of the ladder reaches 8 feet up the wall. How long is the ladder?

Solution: This forms a right-angled triangle. Use the Pythagorean Theorem:

• a = 6, b = 8
• 6² + 8² = c²
• 36 + 64 = c²
• c² = 100
• c = 10

The ladder is 10 feet long.

Problem 8: Proof involving Similar Triangles

Problem: Prove that two triangles are similar using the AA (Angle-Angle) similarity postulate.

Solution: This requires demonstrating that two angles in one triangle are congruent to two angles in the other triangle. The proof involves stating the given information, showing the angle congruences (potentially using properties of parallel lines, transversals, etc.), stating the AA postulate, and concluding that the triangles are similar.

Problem 9: Problem Solving with Isosceles Triangles

Problem: An isosceles triangle has two sides of length 10 and a base of length 12. Find the height of the triangle.

Solution: Draw an altitude from the vertex angle to the midpoint of the base. This creates two right-angled triangles. Use the Pythagorean Theorem on one of them:

• Hypotenuse = 10
• One leg = 12/2 = 6
• Height² + 6² = 10²
• Height² = 100 - 36 = 64
• Height = 8

The height of the triangle is 8.

Problem 10: Challenging Problem combining multiple concepts

Problem: Two triangles share a common angle. One triangle has sides of length 6, 8, and 10. The other has a side of length 12 opposite the common angle. Find the length of the other two sides.

Solution: This problem requires using similar triangles. The first triangle is a 3-4-5 right triangle (it's a multiple of the 3-4-5 triangle). This ratio is crucial. Since the triangles share a common angle, they are similar. Set up a proportion based on the similar triangle ratio:

• 6/x = 8/y = 10/12 (where x and y are the sides we need to find in the second triangle)
• Solving for x: 6/x = 10/12 => x = (6*12)/10 = 7.2
• Solving for y: 8/y = 10/12 => y = (8*12)/10 = 9.6

The lengths of the other two sides of the second triangle are 7.2 and 9.6.

This detailed guide covers a broad range of problems found in a typical Unit 5: Relationships in Triangles Homework 2 assignment. Remember to practice regularly and seek help when needed. Mastering these concepts is a crucial step in your mathematical journey. Always remember to clearly state your reasoning and show all your work for each problem. Good luck!