Math Problem Statement

Find the values of the variables in the given figure involving intersecting lines and alternate interior angles.

Solution

The problem provided involves a figure with intersecting lines, where we need to find the values of the variables xx and yy. Let me break it down step-by-step.

Part A:

We are given the following angles:

  • (8x10)(8x - 10)^\circ
  • (6y+20)(6y + 20)^\circ
  • (7x)(7x)^\circ

Step 1: Identify the relationships between the angles.

From the diagram, we can observe the following angle relationships:

  • (8x10)(8x - 10)^\circ and (7x)(7x)^\circ are alternate interior angles (since the lines are parallel), so they are equal. Therefore: 8x10=7x8x - 10 = 7x Solving this equation for xx: 8x7x=10    x=108x - 7x = 10 \implies x = 10

Step 2: Use the value of xx to find the angles involving yy.

Next, (6y+20)(6y + 20)^\circ and (7x)(7x)^\circ are vertically opposite angles, which are equal. Substituting x=10x = 10 into (7x)(7x)^\circ: 7x=7(10)=707x = 7(10) = 70^\circ Now, set this equal to (6y+20)(6y + 20)^\circ: 6y+20=706y + 20 = 70 Solving for yy: 6y=7020    6y=50    y=506    y=8.336y = 70 - 20 \implies 6y = 50 \implies y = \frac{50}{6} \implies y = 8.33

So, the values are: x=10,y8.33x = 10, \quad y \approx 8.33

Part B:

In this part, you are asked to explain the reasoning.

  • Alternate interior angles were used to equate (8x10)(8x - 10)^\circ and (7x)(7x)^\circ, which gave us the equation 8x10=7x8x - 10 = 7x.
  • Vertically opposite angles were used to equate (6y+20)(6y + 20)^\circ and 7070^\circ to find the value of yy.

Would you like more details on any part of this solution?

Here are five related questions to consider:

  1. What are alternate interior angles, and why are they equal?
  2. How do we determine when two angles are vertically opposite?
  3. If the lines weren't parallel, how would the relationships between the angles change?
  4. What are some other geometric properties of parallel lines and transversals?
  5. How can we use angle relationships to solve more complex geometric problems?

Tip: When solving geometry problems involving parallel lines, always look for angle relationships like alternate interior, corresponding, or vertically opposite angles.

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Math Problem Analysis

Mathematical Concepts

Geometry
Algebra
Angle Relationships
Parallel Lines

Formulas

Alternate Interior Angles are Equal: ∠A = ∠B
Vertically Opposite Angles are Equal

Theorems

Alternate Interior Angles Theorem
Vertically Opposite Angles Theorem

Suitable Grade Level

Grades 7-9