Unit 5: Relationships in Triangles - Homework 2 Answer Key
This answer key provides solutions for Homework 2 in Unit 5, focusing on relationships within triangles. Remember to always show your work and explain your reasoning to receive full credit in your class. This key serves as a guide to check your understanding and identify areas needing further review. Consult your textbook and class notes for additional support.
Note: Since I don't have access to your specific homework assignment, I will provide examples of the types of problems typically found in a Unit 5, Homework 2 assignment on triangle relationships. You can use these examples to check your answers against your actual homework problems. Remember to replace the example values with the values from your assignment.
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 < 13 (False)
- 5 + 13 = 18 > 7 (True)
- 7 + 13 = 20 > 5 (True)
Since one condition of the Triangle Inequality Theorem is not met (5 + 7 < 13), a triangle with these side lengths is not possible.
Section 2: Isosceles and Equilateral Triangles
Isosceles triangles have at least two congruent sides. Equilateral triangles have three congruent sides.
Example Problem 2: Triangle ABC is an isosceles triangle with AB = AC. If angle B = 50°, find the measure of angle C.
Solution:
In an isosceles triangle, the angles opposite the congruent sides are congruent. Therefore, angle C = angle B = 50°.
Example Problem 3: Find the measure of each angle in an equilateral triangle.
Solution:
In an equilateral triangle, all angles are congruent and each angle measures 60° (180°/3 = 60°).
Section 3: Midsegments of Triangles
The midsegment of a triangle is a segment connecting the midpoints of two sides. It is parallel to the third side and half its length.
Example Problem 4: In triangle DEF, M is the midpoint of DE and N is the midpoint of DF. If MN = 8, what is the length of EF?
Solution:
The midsegment MN is parallel to EF and half its length. Therefore, EF = 2 * MN = 2 * 8 = 16.
Section 4: Similar Triangles
Similar triangles have the same shape but different sizes. Their corresponding angles are congruent, and their corresponding sides are proportional.
Example Problem 5: Triangles ABC and DEF are similar. If AB = 6, BC = 8, and DE = 9, find the length of EF.
Solution:
Since the triangles are similar, the ratio of corresponding sides is constant: AB/DE = BC/EF. Therefore, 6/9 = 8/EF. Solving for EF, we get EF = 12.
Remember to always show your work and clearly state your reasoning for each problem. If you have specific questions on a particular problem from your actual homework assignment, please provide the problem details, and I will do my best to assist you.
