Vertical Angles Worksheets - Free Printable

To solve the problems involving vertical angles, we need to use the property that vertical angles are congruent. This means that when two lines intersect, the angles opposite each other are equal.

Let's solve each problem step by step:

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Problem 1:

[Image description: Two intersecting lines with angles labeled as \( x \) and \( 60^\circ \).]

#### Solution:
- The angle labeled \( x \) is a vertical angle to the given \( 60^\circ \) angle.
- Since vertical angles are congruent, \( x = 60^\circ \).

#### Answer:
\[ x = 60^\circ \]

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Problem 2:

[Image description: Two intersecting lines with angles labeled as \( x \), \( 63^\circ \), and \( 47^\circ \).]

#### Solution:
- The angle labeled \( x \) is a vertical angle to the sum of the given angles \( 63^\circ \) and \( 47^\circ \).
- First, calculate the sum of the given angles:
\[
63^\circ + 47^\circ = 110^\circ
\]
- Since vertical angles are congruent, \( x = 110^\circ \).

#### Answer:
\[ x = 110^\circ \]

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Problem 3:

[Image description: Two intersecting lines with angles labeled as \( 2x \), \( 3x - 29^\circ \), and \( 44^\circ \).]

#### Solution:
- The angle labeled \( 2x \) is a vertical angle to the angle labeled \( 3x - 29^\circ \).
- Since vertical angles are congruent, we set up the equation:
\[
2x = 3x - 29
\]
- Solve for \( x \):
\[
2x - 3x = -29 \implies -x = -29 \implies x = 29
\]

#### Answer:
\[ x = 29 \]

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Problem 4:

[Image description: Two intersecting lines with angles labeled as \( (5x - 17)^\circ \), \( (3x + 15)^\circ \), and \( (2x + 18)^\circ \).]

#### Solution:
- The angle labeled \( (5x - 17)^\circ \) is a vertical angle to the angle labeled \( (3x + 15)^\circ \).
- Since vertical angles are congruent, we set up the equation:
\[
5x - 17 = 3x + 15
\]
- Solve for \( x \):
\[
5x - 3x = 15 + 17 \implies 2x = 32 \implies x = 16
\]

#### Answer:
\[ x = 16 \]

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Problem 5:

[Image description: Two intersecting lines with angles labeled as \( (4x + 14)^\circ \), \( (6x - 32)^\circ \), and \( 100^\circ \).]

#### Solution:
- The angle labeled \( (4x + 14)^\circ \) is a vertical angle to the angle labeled \( (6x - 32)^\circ \).
- Since vertical angles are congruent, we set up the equation:
\[
4x + 14 = 6x - 32
\]
- Solve for \( x \):
\[
4x - 6x = -32 - 14 \implies -2x = -46 \implies x = 23
\]

#### Answer:
\[ x = 23 \]

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Problem 6:

[Image description: Two intersecting lines with angles labeled as \( (2x + 35)^\circ \), \( (3x + 10)^\circ \), and \( 115^\circ \).]

#### Solution:
- The angle labeled \( (2x + 35)^\circ \) is a vertical angle to the angle labeled \( (3x + 10)^\circ \).
- Since vertical angles are congruent, we set up the equation:
\[
2x + 35 = 3x + 10
\]
- Solve for \( x \):
\[
2x - 3x = 10 - 35 \implies -x = -25 \implies x = 25
\]

#### Answer:
\[ x = 25 \]

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Final Answers:

1. \( x = 60^\circ \)
2. \( x = 110^\circ \)
3. \( x = 29 \)
4. \( x = 16 \)
5. \( x = 23 \)
6. \( x = 25 \)

\[
\boxed{60, 110, 29, 16, 23, 25}
\]