Math Problem Statement

solve

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

Solution

To solve for xx and zz in the given diagram, we can apply the following geometric principles:

  1. Vertically opposite angles: These angles are equal. In the diagram, the angle labeled (15x91)(15x - 91)^\circ is vertically opposite the angle labeled 106106^\circ. Therefore, they are equal.

    15x91=10615x - 91 = 106

  2. Linear pair: Angles on a straight line add up to 180180^\circ. The angle zz^\circ and the 106106^\circ angle are adjacent angles, forming a linear pair, so:

    z+106=180z + 106 = 180

Step 1: Solve for xx

Using the equation for vertically opposite angles:

15x91=10615x - 91 = 106

Add 9191 to both sides:

15x=106+9115x = 106 + 91 15x=19715x = 197

Now, divide by 1515:

x=1971513.13x = \frac{197}{15} \approx 13.13

Step 2: Solve for zz

Using the equation for the linear pair:

z+106=180z + 106 = 180

Subtract 106106 from both sides:

z=180106z = 180 - 106 z=74z = 74

Final Answers:

  • x13.13x \approx 13.13
  • z=74z = 74

Would you like further details or explanations? Here are five related questions to explore further:

  1. What are supplementary angles, and how do they relate to linear pairs?
  2. How can we use the properties of vertically opposite angles in other geometrical problems?
  3. If another angle is added, forming a triangle, how would you use angle sum properties to find unknown angles?
  4. How does the concept of alternate interior angles apply to parallel lines cut by a transversal?
  5. What would change if the given angles were part of a quadrilateral?

Tip: In geometry, always look for relationships like vertically opposite, supplementary, and complementary angles to simplify angle-based problems!

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Math Problem Analysis

Mathematical Concepts

Geometry
Angle Relationships

Formulas

Linear Pair Postulate: a + b = 180 degrees
Basic Algebraic Manipulation

Theorems

Vertically Opposite Angles are Equal
Linear Pair Angles are Supplementary

Suitable Grade Level

Grades 8-10