Math Problem Statement

In the figure, lines m and n are parallel. If x = 6k + 13 and y = 8k - 29, what is the value of z?

Solution

Let’s analyze the problem.

Given:

  • Lines mm and nn are parallel.
  • x=6k+13x = 6k + 13 and y=8k29y = 8k - 29.
  • The goal is to find the value of zz.

Key Observations:

  1. Since mm and nn are parallel, xx and yy are alternate interior angles. This means x=yx = y.
  2. Solving x=yx = y allows us to find kk, and then substitute to determine zz.

Steps:

  1. Set x=yx = y: 6k+13=8k296k + 13 = 8k - 29

  2. Solve for kk: 13+29=8k6k13 + 29 = 8k - 6k 42=2k    k=2142 = 2k \implies k = 21

  3. Using k=21k = 21, substitute into zz's formula. Based on parallel line geometry, zz is supplementary to xx (since zz and xx form a linear pair): z=180xz = 180^\circ - x Substitute k=21k = 21 into x=6k+13x = 6k + 13: x=6(21)+13=126+13=139x = 6(21) + 13 = 126 + 13 = 139 Now calculate zz: z=180139=41z = 180 - 139 = 41

Final Answer:

41\boxed{41}

Would you like me to elaborate on any of these steps or discuss related topics?


Here are 5 related questions to explore further:

  1. What is the definition of alternate interior angles, and how do they relate to parallel lines?
  2. How do you calculate supplementary angles in geometric diagrams?
  3. How can algebra be used to solve for unknown variables in geometric contexts?
  4. What other angle relationships exist when parallel lines are intersected by a transversal?
  5. How does changing the values of kk affect the angles in this figure?

Tip: Always identify and use angle relationships like supplementary, complementary, and vertical angles when solving geometry problems!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Parallel Lines
Angles
Algebra

Formulas

x = y (Alternate Interior Angles)
z = 180 - x (Supplementary Angles)

Theorems

Alternate Interior Angle Theorem
Supplementary Angle Theorem

Suitable Grade Level

Grades 8-10