Angle pairs worksheet | TPT - Free Printable

Problem Analysis:

The task involves solving for the measures of various angles in a geometric diagram and classifying them based on their relationships. The given information includes:

1. Given Angles:
- \( m\angle 2 = 65^\circ \)
- \( m\angle 3 = 2x + 45^\circ \)
- \( m\angle 7 = 90^\circ \)
- \( m\angle 8 = x \)

2. Tasks:
- Solve for the measure of each angle.
- Classify the angles as supplementary, complementary, adjacent, or vertical.

---

Step-by-Step Solution:

#### 1. Solve for \( x \) using the given angles:
From the diagram, we observe that \( \angle 3 \) and \( \angle 8 \) are vertically opposite angles. Vertically opposite angles are always equal. Therefore:
\[
m\angle 3 = m\angle 8
\]
Substitute the given expressions:
\[
2x + 45^\circ = x
\]
Solve for \( x \):
\[
2x + 45^\circ = x \implies 2x - x = -45^\circ \implies x = 45^\circ
\]

#### 2. Calculate the measures of all angles:
Using \( x = 45^\circ \), we can now find the measures of all angles.

- \( m\angle 8 \):
\[
m\angle 8 = x = 45^\circ
\]

- \( m\angle 3 \):
\[
m\angle 3 = 2x + 45^\circ = 2(45^\circ) + 45^\circ = 90^\circ + 45^\circ = 135^\circ
\]

- Other Angles:
Using the properties of angles formed by intersecting lines and parallel lines (if applicable), we can determine the measures of the remaining angles. However, the problem does not explicitly mention parallel lines, so we assume standard angle relationships (e.g., supplementary, complementary, adjacent, vertical).

#### 3. Classify the angles:
We need to classify the angles as supplementary, complementary, adjacent, or vertical. Let's analyze the relationships:

- Supplementary Angles: Two angles are supplementary if their measures add up to \( 180^\circ \).
- Complementary Angles: Two angles are complementary if their measures add up to \( 90^\circ \).
- Adjacent Angles: Two angles are adjacent if they share a common vertex and a common side.
- Vertical Angles: Two angles are vertical if they are opposite each other at an intersection of two lines.

From the diagram:
- Supplementary Angles:
- \( \angle 3 \) and \( \angle 4 \) are supplementary because they form a linear pair.
- \( \angle 6 \) and \( \angle 7 \) are supplementary because they form a linear pair.
- \( \angle 1 \) and \( \angle 2 \) are supplementary because they form a linear pair.
- \( \angle 5 \) and \( \angle 8 \) are supplementary because they form a linear pair.

- Complementary Angles:
- None of the angles in the diagram are complementary because none of them add up to \( 90^\circ \).

- Adjacent Angles:
- \( \angle 1 \) and \( \angle 2 \) are adjacent.
- \( \angle 2 \) and \( \angle 3 \) are adjacent.
- \( \angle 3 \) and \( \angle 4 \) are adjacent.
- \( \angle 4 \) and \( \angle 1 \) are adjacent.
- \( \angle 5 \) and \( \angle 6 \) are adjacent.
- \( \angle 6 \) and \( \angle 7 \) are adjacent.
- \( \angle 7 \) and \( \angle 8 \) are adjacent.
- \( \angle 8 \) and \( \angle 5 \) are adjacent.

- Vertical Angles:
- \( \angle 1 \) and \( \angle 3 \) are vertical angles.
- \( \angle 2 \) and \( \angle 4 \) are vertical angles.
- \( \angle 5 \) and \( \angle 7 \) are vertical angles.
- \( \angle 6 \) and \( \angle 8 \) are vertical angles.

#### 4. Fill in the chart:
Using the classifications above, we can fill in the chart:

| Classification | Angle Pairs |
|----------------------|------------------------------|
| Supplementary | \( \angle 3 \) and \( \angle 4 \) |
| | \( \angle 6 \) and \( \angle 7 \) |
| | \( \angle 1 \) and \( \angle 2 \) |
| | \( \angle 5 \) and \( \angle 8 \) |
| Complementary | None |
| Adjacent | \( \angle 1 \) and \( \angle 2 \) |
| | \( \angle 2 \) and \( \angle 3 \) |
| | \( \angle 3 \) and \( \angle 4 \) |
| | \( \angle 4 \) and \( \angle 1 \) |
| | \( \angle 5 \) and \( \angle 6 \) |
| | \( \angle 6 \) and \( \angle 7 \) |
| | \( \angle 7 \) and \( \angle 8 \) |
| | \( \angle 8 \) and \( \angle 5 \) |
| Vertical | \( \angle 1 \) and \( \angle 3 \) |
| | \( \angle 2 \) and \( \angle 4 \) |
| | \( \angle 5 \) and \( \angle 7 \) |
| | \( \angle 6 \) and \( \angle 8 \) |

---

Final Answer:

\[
\boxed{
\begin{array}{|c|c|}
\hline
\text{Classification} & \text{Angle Pairs} \\
\hline
\text{Supplementary} & \angle 3 \text{ and } \angle 4, \angle 6 \text{ and } \angle 7, \angle 1 \text{ and } \angle 2, \angle 5 \text{ and } \angle 8 \\
\hline
\text{Complementary} & \text{None} \\
\hline
\text{Adjacent} & \angle 1 \text{ and } \angle 2, \angle 2 \text{ and } \angle 3, \angle 3 \text{ and } \angle 4, \angle 4 \text{ and } \angle 1, \angle 5 \text{ and } \angle 6, \angle 6 \text{ and } \angle 7, \angle 7 \text{ and } \angle 8, \angle 8 \text{ and } \angle 5 \\
\hline
\text{Vertical} & \angle 1 \text{ and } \angle 3, \angle 2 \text{ and } \angle 4, \angle 5 \text{ and } \angle 7, \angle 6 \text{ and } \angle 8 \\
\hline
\end{array}
}
\]