Question

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!