https://file.aiquickdraw.com/mathbot/file/1726801511890yfskpnqm.jpg

The image shows two intersecting lines forming a pair of opposite (or vertical) angles. One pair of opposite angles is labeled as \(110^\circ\), while the expressions for the other pair of opposite angles are given as algebraic expressions:

1. \(5y + 10\)
2. \(6y - 2\)
3. \(2x + 40\)

### Step-by-step analysis:

#### Vertical Angles Theorem:
According to the Vertical Angles Theorem, opposite angles formed by intersecting lines are equal. Therefore, the following equalities must hold:

- \(5y + 10 = 110\) (because the angle opposite \(5y + 10\) is \(110^\circ\))
- \(6y - 2 = 110\) (because the angle opposite \(6y - 2\) is also \(110^\circ\))

### Solving for \(y\):

#### 1. \(5y + 10 = 110\):
\[
5y + 10 = 110
\]
\[
5y = 110 - 10
\]
\[
5y = 100
\]
\[
y = \frac{100}{5} = 20
\]

#### 2. \(6y - 2 = 110\):
Substitute \(y = 20\) into this equation to check consistency:
\[
6(20) - 2 = 110
\]
\[
120 - 2 = 110
\]
This confirms that \(y = 20\) is correct.

### Next:
For the expression \(2x + 40\), the corresponding angle should also be \(110^\circ\), so we can set:
\[
2x + 40 = 110
\]

#### Solving for \(x\):
\[
2x + 40 = 110
\]
\[
2x = 110 - 40
\]
\[
2x = 70
\]
\[
x = \frac{70}{2} = 35
\]

### Final Answers:
- \(y = 20\)
- \(x = 35\)

Would you like further details on this? Here are some related questions you might consider:

1. What are vertical angles and why are they always equal?
2. How do you solve more complex equations involving multiple variables?
3. What are supplementary angles, and how do they relate to this type of problem?
4. Can intersecting lines form other types of angle relationships?
5. How does the solution change if the lines are not straight?

**Tip**: When solving for variables in angle equations, always double-check consistency across all given expressions!

The image shows intersecting lines with the following expressions: 5y + 10, 6y - 2, and 2x + 40, along with an angle labeled 110 degrees.

This problem involves solving for variables using the Vertical Angles Theorem. With two intersecting lines, one angle is given as 110°, while other angles are expressed algebraically. The solution involves solving for x and y using algebraic equations. Suitable for students in Grades 7-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation