Solve equations with vertical angles practice

Answer 1

Vertical angles are angles that are opposite each other when two lines intersect. When two angles are vertical angles, they are equal in measure.

Here are a few examples of equations involving vertical angles:

Example 1

Problem:
Line 1 intersects Line 2, creating two pairs of vertical angles. One angle measures \( 3x + 10 \) degrees, and the vertical angle measures \( 5x - 2 \) degrees. Find \( x \).

Solution: Set the two angles equal to each other: \[ 3x + 10 = 5x - 2 \]

Now, solve for \( x \):

1. Subtract \( 3x \) from both sides: \[ 10 = 2x - 2 \]

2. Add \( 2 \) to both sides: \[ 12 = 2x \]

3. Divide by \( 2 \): \[ x = 6 \]

Example 2

Problem:
Two intersecting lines create angles of \( 2y + 15 \) degrees and \( 4y - 7 \) degrees as vertical angles. Find \( y \).

Solution: Set the angles equal: \[ 2y + 15 = 4y - 7 \]

Now, solve for \( y \):

1. Subtract \( 2y \) from both sides: \[ 15 = 2y - 7 \]

2. Add \( 7 \) to both sides: \[ 22 = 2y \]

3. Divide by \( 2 \): \[ y = 11 \]

Example 3

Problem:
The measures of two vertical angles are given by \( 8a + 4 \) and \( 6a + 28 \). Find \( a \).

Solution: Set the angles equal to each other: \[ 8a + 4 = 6a + 28 \]

Now, solve for \( a \):

1. Subtract \( 6a \) from both sides: \[ 2a + 4 = 28 \]

2. Subtract \( 4 \) from both sides: \[ 2a = 24 \]

3. Divide by \( 2 \): \[ a = 12 \]

Practice Problems

Now, you can try solving these equations involving vertical angles:

1. \( 5x + 8 = 7x - 10 \)
2. \( 4y - 5 = 2y + 15 \)
3. \( 12a + 3 = 10a + 27 \)

Solutions

1. For \( 5x + 8 = 7x - 10 \):

   • \( 18 = 2x \), so \( x = 9 \).

2. For \( 4y - 5 = 2y + 15 \):

   • \( 2y = 20 \), so \( y = 10 \).

3. For \( 12a + 3 = 10a + 27 \):

   • \( 2a = 24 \), so \( a = 12 \).

Feel free to reach out if you have any questions or need help with more problems!