Vertical Angles Worksheets (printable, online, answers, examples) - Free Printable

Problem Analysis and Solution

The worksheet involves solving for unknown variables using the properties of vertical angles and linear pairs. Let's solve each part step by step.

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#### Part 1: Finding the value of \( x \) in the first figure

Given:
- \( PQ \) and \( RS \) are straight lines.
- One angle is given as \( 130^\circ \).
- The other angle involving \( x \) is labeled as \( 2x \).

Solution:
When two lines intersect, the angles opposite each other (called vertical angles) are equal. In this case:
- The angle \( 2x \) is vertically opposite to the angle \( 130^\circ \).

Thus, we can write:
\[
2x = 130^\circ
\]

Solving for \( x \):
\[
x = \frac{130^\circ}{2} = 65^\circ
\]

Answer for Part 1:
\[
\boxed{65}
\]

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#### Part 2: Finding the value of \( x \) in the second figure

Given:
- \( PQ \) and \( RS \) are straight lines.
- One angle is given as \( 120^\circ \).
- Another angle is given as \( 150^\circ \).
- The angle involving \( x \) is labeled as \( x \).

Solution:
In this figure, the angle \( x \) is part of a linear pair with the angle \( 150^\circ \). A linear pair means the two angles add up to \( 180^\circ \). Therefore:
\[
x + 150^\circ = 180^\circ
\]

Solving for \( x \):
\[
x = 180^\circ - 150^\circ = 30^\circ
\]

Answer for Part 2:
\[
\boxed{30}
\]

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#### Part 3: Finding the values of \( x \), \( y \), and \( z \) in the third figure

Given:
- \( AB \) and \( CD \) are straight lines.
- One angle is given as \( 140^\circ \).
- Other angles are labeled as \( 2x \), \( 4z \), and \( y \).

Solution:
1. Finding \( y \):
- The angle \( y \) is vertically opposite to the angle \( 140^\circ \). Since vertical angles are equal:
\[
y = 140^\circ
\]

2. Finding \( 2x \):
- The angle \( 2x \) is part of a linear pair with the angle \( 140^\circ \). Therefore:
\[
2x + 140^\circ = 180^\circ
\]
- Solving for \( 2x \):
\[
2x = 180^\circ - 140^\circ = 40^\circ
\]
- Solving for \( x \):
\[
x = \frac{40^\circ}{2} = 20^\circ
\]

3. Finding \( 4z \):
- The angle \( 4z \) is vertically opposite to the angle \( 2x \). Since vertical angles are equal:
\[
4z = 2x
\]
- Substituting \( 2x = 40^\circ \):
\[
4z = 40^\circ
\]
- Solving for \( z \):
\[
z = \frac{40^\circ}{4} = 10^\circ
\]

Answers for Part 3:
\[
x = 20, \quad y = 140, \quad z = 10
\]
\[
\boxed{20, 140, 10}
\]

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#### Part 4: Finding the sizes of the other three angles when one angle is \( 127^\circ \)

Given:
- One of the angles formed when two lines intersect is \( 127^\circ \).

Solution:
When two lines intersect, four angles are formed. These angles have the following properties:
1. Vertical angles are equal.
2. Angles forming a linear pair add up to \( 180^\circ \).

Let the four angles be \( A, B, C, \) and \( D \), where:
- \( A = 127^\circ \)
- \( B \) is the angle vertically opposite to \( A \).
- \( C \) and \( D \) are the angles forming linear pairs with \( A \) and \( B \), respectively.

1. Finding \( B \):
- Since \( B \) is vertically opposite to \( A \):
\[
B = 127^\circ
\]

2. Finding \( C \):
- \( C \) forms a linear pair with \( A \):
\[
C + 127^\circ = 180^\circ
\]
\[
C = 180^\circ - 127^\circ = 53^\circ
\]

3. Finding \( D \):
- \( D \) is vertically opposite to \( C \):
\[
D = 53^\circ
\]

Sizes of the other three angles:
\[
\boxed{127^\circ, 53^\circ, 53^\circ}
\]

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Final Answers:

1. \( \boxed{65} \)
2. \( \boxed{30} \)
3. \( \boxed{20, 140, 10} \)
4. \( \boxed{127^\circ, 53^\circ, 53^\circ} \)