Vertical Angles Worksheets - Free Printable

To solve the problem, we need to analyze the given geometric figure and use the properties of angles formed by intersecting lines. Let's break it down step by step.

Step 1: Understand the Problem

We are given a diagram with several intersecting lines forming various angles. The task is to determine which of the given equations are true based on the properties of vertical angles and supplementary angles.

Step 2: Recall Key Angle Properties

1. Vertical Angles: When two lines intersect, the angles opposite each other are called vertical angles. Vertical angles are always congruent.
- If two angles are vertical angles, then their measures are equal.

2. Supplementary Angles: Two angles are supplementary if their measures add up to 180 degrees.
- If two angles form a linear pair (i.e., they are adjacent and lie on a straight line), then they are supplementary.

Step 3: Analyze the Diagram

From the diagram:
- Angles $ \angle AOC $ and $ \angle BOD $ are vertical angles.
- Angles $ \angle AOB $ and $ \angle BOC $ are adjacent angles forming a linear pair.
- Angles $ \angle BOC $ and $ \angle COD $ are adjacent angles forming a linear pair.
- Angles $ \angle AOD $ and $ \angle BOC $ are also related through the properties of intersecting lines.

Step 4: Evaluate Each Equation

#### Equation I: $ m\angle AOC = m\angle BOD $
- These are vertical angles. By the property of vertical angles, they are congruent.
- Therefore, $ m\angle AOC = m\angle BOD $ is true.

#### Equation II: $ m\angle AOB + m\angle BOC = 180^\circ $
- Angles $ \angle AOB $ and $ \angle BOC $ form a linear pair.
- By the property of supplementary angles, their measures add up to 180 degrees.
- Therefore, $ m\angle AOB + m\angle BOC = 180^\circ $ is true.

#### Equation III: $ m\angle AOD = m\angle BOC $
- These angles are not vertical angles, nor are they directly related as supplementary angles from the given information.
- Without additional information, we cannot conclude that $ m\angle AOD = m\angle BOC $.
- Therefore, $ m\angle AOD = m\angle BOC $ is false.

Step 5: Determine the Correct Answer

From the analysis:
- Equation I is true.
- Equation II is true.
- Equation III is false.

Thus, the correct answer is:
$$
\boxed{\text{I and II}}
$$