Math Problem Statement

In the figure below, o || n. Find the values of x and z. The given angles are (6z - 95)°, x°, and 77°.

Solution

In this problem, you are asked to find the values of xx and zz based on the figure, where lines ono \parallel n, meaning the two lines are parallel, and you are given the following angles:

  • (6z95)(6z - 95)^\circ on the top left.
  • xx^\circ on the bottom left.
  • 7777^\circ on the bottom right.

Step 1: Using the parallel line properties

Since ono \parallel n, we can apply the properties of alternate interior angles and corresponding angles.

  1. Corresponding angles: The angle marked 7777^\circ corresponds to the angle xx^\circ on the left because they are on the same side of the transversal and in corresponding positions. Therefore, we have: x=77x = 77^\circ

Step 2: Solving for zz

The angles (6z95)(6z - 95)^\circ and 7777^\circ are alternate interior angles, which are equal because the lines are parallel. Thus, we set the two angles equal: 6z95=776z - 95 = 77

Step 3: Solving the equation for zz

Now, solve the equation: 6z95=776z - 95 = 77 Add 95 to both sides: 6z=1726z = 172 Now divide by 6: z=1726=28.67z = \frac{172}{6} = 28.67

Final answers:

  • x=77x = 77^\circ
  • z28.67z \approx 28.67

Would you like any additional details or clarification on the steps?

Here are 5 related questions for further exploration:

  1. What are alternate interior angles, and why are they equal when lines are parallel?
  2. How do corresponding angles help in solving geometry problems with parallel lines?
  3. How would the solution change if the lines were not parallel?
  4. Can you explain how vertical angles work in relation to parallel lines and a transversal?
  5. What are the different types of angles formed when a transversal crosses two parallel lines?

Tip: When solving geometric problems involving parallel lines, always look for alternate interior and corresponding angles, as they often provide direct relationships.

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Math Problem Analysis

Mathematical Concepts

Parallel Lines
Transversals
Corresponding Angles
Alternate Interior Angles

Formulas

x = 77° (Corresponding Angles)
6z - 95 = 77 (Alternate Interior Angles)

Theorems

Corresponding Angles Theorem
Alternate Interior Angles Theorem

Suitable Grade Level

Grades 8-10