Unit 5: Relationships in Triangles - Homework 5 Answer Key
This answer key provides solutions for Homework 5 in Unit 5, focusing on relationships within triangles. Remember to always show your work and explain your reasoning, as this demonstrates a deeper understanding of the concepts. This key provides the answers, but focusing on the process is crucial for mastering the material.
Note: Since I do not have access to the specific questions in your Homework 5 assignment, I will provide example problems and solutions covering common topics within this unit. You can use these examples as a guide to check your own work.
Section 1: Triangle Inequality Theorem
The Triangle Inequality 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.
Example Problem 1: Can a triangle have sides with lengths 5, 7, and 13?
Solution:
- 5 + 7 = 12, which is less than 13.
- Therefore, a triangle with these side lengths is not possible. The Triangle Inequality Theorem is violated.
Example Problem 2: Find the range of possible values for the third side of a triangle with sides of length 4 and 9.
Solution:
Let 'x' be the length of the third side. According to the Triangle Inequality Theorem:
- x + 4 > 9 => x > 5
- x + 9 > 4 => x > -5 (This is always true since side lengths are positive)
- 4 + 9 > x => 13 > x
Therefore, the range of possible values for x is 5 < x < 13.
Section 2: Similar Triangles
Similar triangles have the same shape, but not necessarily the same size. Corresponding angles are congruent, and corresponding sides are proportional.
Example Problem 3: Two triangles are similar. The sides of the first triangle are 3, 4, and 5. The shortest side of the second triangle is 9. Find the lengths of the other two sides.
Solution:
Since the triangles are similar, the ratio of corresponding sides is constant. The shortest side of the first triangle is 3, and the shortest side of the second triangle is 9. The ratio is 9/3 = 3. Therefore, we multiply the other sides of the first triangle by 3:
- 4 * 3 = 12
- 5 * 3 = 15
The lengths of the other two sides of the second triangle are 12 and 15.
Section 3: Isosceles and Equilateral Triangles
- Isosceles Triangles: Have at least two congruent sides.
- Equilateral Triangles: Have three congruent sides (and three congruent angles of 60° each).
Example Problem 4: In an isosceles triangle, two angles measure 50° and 80°. What is the measure of the third angle?
Solution:
Since the triangle is isosceles, it must have at least two congruent angles. The angles must add up to 180°. The possible combinations are:
- 50° + 50° + 80° = 180° (This is possible)
- 50° + 80° + 50° = 180° (This is possible)
Therefore, the third angle can be either 50° or 80°.
Remember to consult your textbook and class notes for specific problems and formulas relevant to your Homework 5 assignment. This answer key is intended as a general guide, not a direct replacement for your own work.
