Math Problem Statement

Given the figure below, find the values of x and z.

Solution

To solve for xx and zz in this figure, we will use the properties of vertically opposite angles.

  1. The expressions (10x+68)(10x + 68)^\circ and (13x+53)(13x + 53)^\circ represent two vertically opposite angles, which are equal. Thus, we set up the equation: 10x+68=13x+5310x + 68 = 13x + 53

  2. Rearrange to solve for xx: 6853=13x10x68 - 53 = 13x - 10x 15=3x15 = 3x x=5x = 5

  3. Now, substitute x=5x = 5 back into one of the angle expressions (for example, 10x+6810x + 68) to find the angle value: 10(5)+68=50+68=11810(5) + 68 = 50 + 68 = 118^\circ

  4. Since vertically opposite angles are equal, both (10x+68)(10x + 68)^\circ and (13x+53)(13x + 53)^\circ equal 118118^\circ.

  5. For the value of zz, observe that zz^\circ is the supplementary angle to 118118^\circ (since they form a straight line), so: z=180118=62z = 180^\circ - 118^\circ = 62^\circ

Final answers:

  • x=5x = 5
  • z=62z = 62^\circ

Would you like further details or have any questions?

Here are some related questions for additional practice:

  1. What is the sum of angles around a point?
  2. How do you identify vertically opposite angles?
  3. How can you solve for unknowns in equations involving angles?
  4. What are supplementary angles, and how are they related?
  5. How does rearranging equations help in solving for unknowns?

Tip: Always remember that vertically opposite angles are equal. This property is particularly useful when working with intersecting lines.

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Math Problem Analysis

Mathematical Concepts

Algebra
Geometry
Vertically Opposite Angles
Supplementary Angles

Formulas

Vertically opposite angles are equal: (10x + 68)° = (13x + 53)°
Supplementary angles: z = 180° - 118°

Theorems

Vertically Opposite Angles Theorem
Supplementary Angles Theorem

Suitable Grade Level

Grades 8-10