Math Problem Statement

In the figure below, m || n. Find the values of z and x.

Solution

In this figure, we are given that the lines mnm \parallel n, and we need to find the values of zz and xx. Let's proceed step-by-step using the properties of parallel lines and transversals.

Step 1: Find zz

Since mnm \parallel n, the angles formed by the transversal, such as the 66° and zz, are corresponding angles. Corresponding angles are equal when lines are parallel. Hence, we have:

z=66z = 66^\circ

Step 2: Find xx

For the second part, we have a linear equation involving the angle expression (5x99)(5x - 99)^\circ, which is vertically opposite to the zz^\circ angle (since vertical angles are equal). Therefore, we set:

5x99=z5x - 99 = z

Substituting z=66z = 66^\circ from the first step:

5x99=665x - 99 = 66

Now, solve for xx:

5x=66+995x = 66 + 99 5x=1655x = 165 x=1655=33x = \frac{165}{5} = 33

Final Answer:

  • z=66z = 66^\circ
  • x=33x = 33

Would you like more details or further clarification on any of the steps?

Here are five additional questions to deepen your understanding:

  1. What are alternate interior angles, and how are they different from corresponding angles?
  2. How can the property of vertical angles help in solving geometry problems?
  3. What would happen to the values of xx and zz if the lines mm and nn were not parallel?
  4. How would you solve for xx and zz if zz had a different value than 66°?
  5. Can we apply the same method if there are more than two parallel lines? Why?

Tip: Always look for angle relationships like corresponding, alternate, and vertical angles when working with parallel lines and transversals.

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Math Problem Analysis

Mathematical Concepts

Geometry
Parallel Lines
Angles

Formulas

Corresponding angles are equal when lines are parallel
Vertical angles are equal

Theorems

Corresponding Angles Postulate
Vertical Angles Theorem

Suitable Grade Level

Grades 8-10