Main - Math 8 Classify Angles Worksheet | PDF | Angle | Elementary ... - Free Printable

Problem Analysis and Solution

The task involves classifying angles as complementary, supplementary, or adjacent, and finding the value of \( x \) in each figure. Let's solve each part step by step.

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#### 1. Classify angles and find \( x \)

##### Figure 1
- Given: Two angles forming a straight line: \( 120^\circ \) and \( x^\circ \).
- Classification: These angles are supplementary because they form a straight line.
- Reasoning: The sum of angles on a straight line is \( 180^\circ \).
- Equation: \( 120^\circ + x^\circ = 180^\circ \)
- Solution:
\[
x = 180^\circ - 120^\circ = 60^\circ
\]
- Answer: Supplementary, \( x = 60^\circ \).

##### Figure 2
- Given: Two vertical angles: \( 119^\circ \) and \( x^\circ \).
- Classification: These angles are vertical angles.
- Reasoning: Vertical angles are always equal.
- Equation: \( x^\circ = 119^\circ \)
- Solution:
\[
x = 119^\circ
\]
- Answer: Vertical, \( x = 119^\circ \).

##### Figure 3
- Given: Two angles forming a right angle: \( x^\circ \) and \( 55^\circ \).
- Classification: These angles are complementary because they form a right angle.
- Reasoning: The sum of complementary angles is \( 90^\circ \).
- Equation: \( x^\circ + 55^\circ = 90^\circ \)
- Solution:
\[
x = 90^\circ - 55^\circ = 35^\circ
\]
- Answer: Complementary, \( x = 35^\circ \).

##### Figure 4
- Given: Two angles forming a straight line: \( 40^\circ \) and \( x^\circ \).
- Classification: These angles are supplementary because they form a straight line.
- Reasoning: The sum of angles on a straight line is \( 180^\circ \).
- Equation: \( 40^\circ + x^\circ = 180^\circ \)
- Solution:
\[
x = 180^\circ - 40^\circ = 140^\circ
\]
- Answer: Supplementary, \( x = 140^\circ \).

##### Figure 5
- Given: A right triangle with one angle \( 80^\circ \) and another angle \( x^\circ \).
- Classification: These angles are part of a triangle.
- Reasoning: The sum of angles in a triangle is \( 180^\circ \). One angle is \( 90^\circ \) (right angle), so:
\[
80^\circ + x^\circ + 90^\circ = 180^\circ
\]
- Equation: \( 80^\circ + x^\circ + 90^\circ = 180^\circ \)
- Solution:
\[
x = 180^\circ - 80^\circ - 90^\circ = 10^\circ
\]
- Answer: Triangle angles, \( x = 10^\circ \).

##### Figure 6
- Given: Two angles forming a straight line: \( 98^\circ \) and \( x^\circ \).
- Classification: These angles are supplementary because they form a straight line.
- Reasoning: The sum of angles on a straight line is \( 180^\circ \).
- Equation: \( 98^\circ + x^\circ = 180^\circ \)
- Solution:
\[
x = 180^\circ - 98^\circ = 82^\circ
\]
- Answer: Supplementary, \( x = 82^\circ \).

##### Figure 7
- Given: Two angles forming a right angle: \( 22^\circ \) and \( x^\circ \).
- Classification: These angles are complementary because they form a right angle.
- Reasoning: The sum of complementary angles is \( 90^\circ \).
- Equation: \( 22^\circ + x^\circ = 90^\circ \)
- Solution:
\[
x = 90^\circ - 22^\circ = 68^\circ
\]
- Answer: Complementary, \( x = 68^\circ \).

##### Figure 8
- Given: Two vertical angles: \( 59^\circ \) and \( x^\circ \).
- Classification: These angles are vertical angles.
- Reasoning: Vertical angles are always equal.
- Equation: \( x^\circ = 59^\circ \)
- Solution:
\[
x = 59^\circ
\]
- Answer: Vertical, \( x = 59^\circ \).

##### Figure 9
- Given: Two angles forming a straight line: \( x^\circ \) and \( 6^\circ \).
- Classification: These angles are supplementary because they form a straight line.
- Reasoning: The sum of angles on a straight line is \( 180^\circ \).
- Equation: \( x^\circ + 6^\circ = 180^\circ \)
- Solution:
\[
x = 180^\circ - 6^\circ = 174^\circ
\]
- Answer: Supplementary, \( x = 174^\circ \).

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#### 10. Find the measure of angles 1, 2, and 3

- Given: Angles 1, 2, and 3 are adjacent angles on a straight line, with \( \angle 3 = 43^\circ \).
- Reasoning: Angles on a straight line sum to \( 180^\circ \). Since \( \angle 2 \) and \( \angle 3 \) are adjacent and form a straight line:
\[
\angle 2 + \angle 3 = 180^\circ
\]
Also, \( \angle 1 \) and \( \angle 2 \) are adjacent and form a straight line:
\[
\angle 1 + \angle 2 = 180^\circ
\]
- Solution:
\[
\angle 2 = 180^\circ - \angle 3 = 180^\circ - 43^\circ = 137^\circ
\]
\[
\angle 1 = 180^\circ - \angle 2 = 180^\circ - 137^\circ = 43^\circ
\]
- Answer: \( \angle 1 = 43^\circ \), \( \angle 2 = 137^\circ \), \( \angle 3 = 43^\circ \).

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#### 11. Name the angles

- Given: A diagram with labeled angles \( a, b, c, d, e \).

##### a) Vertical angles
- Definition: Vertical angles are opposite angles formed by two intersecting lines.
- Identified: \( \angle a \) and \( \angle d \), \( \angle b \) and \( \angle e \).
- Answer: Vertical: \( \angle a \) and \( \angle d \), \( \angle b \) and \( \angle e \).

##### b) Complementary angles
- Definition: Complementary angles sum to \( 90^\circ \).
- Identified: \( \angle c \) and \( \angle e \) (since \( \angle c \) and \( \angle e \) form a right angle).
- Answer: Complementary: \( \angle c \) and \( \angle e \).

##### c) Supplementary angles
- Definition: Supplementary angles sum to \( 180^\circ \).
- Identified: \( \angle c \) and \( \angle d \) (since they form a straight line).
- Answer: Supplementary: \( \angle c \) and \( \angle d \).

##### d) All adjacent angles
- Definition: Adjacent angles share a common vertex and side.
- Identified: \( \angle a \) and \( \angle b \), \( \angle b \) and \( \angle c \), \( \angle c \) and \( \angle d \), \( \angle d \) and \( \angle e \), \( \angle e \) and \( \angle a \).
- Answer: All adjacent: \( \angle a \) and \( \angle b \), \( \angle b \) and \( \angle c \), \( \angle c \) and \( \angle d \), \( \angle d \) and \( \angle e \), \( \angle e \) and \( \angle a \).

##### e) Find values of all angles if \( \angle c = 30^\circ \)
- Given: \( \angle c = 30^\circ \).
- Reasoning:
1. \( \angle c \) and \( \angle d \) are supplementary:
\[
\angle d = 180^\circ - \angle c = 180^\circ - 30^\circ = 150^\circ
\]
2. \( \angle c \) and \( \angle e \) are complementary:
\[
\angle e = 90^\circ - \angle c = 90^\circ - 30^\circ = 60^\circ
\]
3. \( \angle a \) and \( \angle d \) are vertical:
\[
\angle a = \angle d = 150^\circ
\]
4. \( \angle b \) and \( \angle e \) are vertical:
\[
\angle b = \angle e = 60^\circ
\]
- Answer: \( \angle a = 150^\circ \), \( \angle b = 60^\circ \), \( \angle c = 30^\circ \), \( \angle d = 150^\circ \), \( \angle e = 60^\circ \).

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Final Answers

\[
\boxed{
\begin{array}{ll}
1. & \text{Supplementary, } x = 60^\circ \\
2. & \text{Vertical, } x = 119^\circ \\
3. & \text{Complementary, } x = 35^\circ \\
4. & \text{Supplementary, } x = 140^\circ \\
5. & \text{Triangle angles, } x = 10^\circ \\
6. & \text{Supplementary, } x = 82^\circ \\
7. & \text{Complementary, } x = 68^\circ \\
8. & \text{Vertical, } x = 59^\circ \\
9. & \text{Supplementary, } x = 174^\circ \\
10. & \angle 1 = 43^\circ, \angle 2 = 137^\circ, \angle 3 = 43^\circ \\
11. & \text{(a) Vertical: } \angle a \text{ and } \angle d, \angle b \text{ and } \angle e \\
& \text{(b) Complementary: } \angle c \text{ and } \angle e \\
& \text{(c) Supplementary: } \angle c \text{ and } \angle d \\
& \text{(d) All adjacent: } \angle a \text{ and } \angle b, \angle b \text{ and } \angle c, \angle c \text{ and } \angle d, \angle d \text{ and } \angle e, \angle e \text{ and } \angle a \\
& \text{(e) } \angle a = 150^\circ, \angle b = 60^\circ, \angle c = 30^\circ, \angle d = 150^\circ, \angle e = 60^\circ
\end{array}
}
\]