Explanation of vertical angles and their properties when two lines intersect.

Problem Analysis:

The task involves understanding and applying the concept of vertical angles. The key points from the image are:

1. Definition of Vertical Angles: When two lines intersect, they form four angles. The angles opposite each other (across the intersection point) are called vertical angles.
2. Property of Vertical Angles: Vertical angles are always equal in measure.
3. Supplementary Angles: Adjacent angles formed by the intersection are supplementary, meaning their measures add up to 180°.

The goal is to use these properties to solve for unknown angles in a given scenario.

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Solution Explanation:

#### Step 1: Understand the Property of Vertical Angles
From the image, we see that vertical angles are always equal. For example:
- If two lines intersect, and one pair of vertical angles is \(a\) and \(c\), then \(a = c\).
- Similarly, if another pair of vertical angles is \(b\) and \(d\), then \(b = d\).

#### Step 2: Use the Supplementary Angle Property
Adjacent angles formed by the intersection of two lines are supplementary. This means:
- If two adjacent angles are \(a\) and \(b\), then \(a + b = 180^\circ\).
- Similarly, if two other adjacent angles are \(c\) and \(d\), then \(c + d = 180^\circ\).

#### Step 3: Apply the Properties to Solve for Unknowns
Let's consider the general case where we have two intersecting lines forming four angles: \(a\), \(b\), \(c\), and \(d\). According to the properties:
1. \(a\) and \(c\) are vertical angles, so \(a = c\).
2. \(b\) and \(d\) are vertical angles, so \(b = d\).
3. \(a\) and \(b\) are supplementary, so \(a + b = 180^\circ\).
4. \(c\) and \(d\) are supplementary, so \(c + d = 180^\circ\).

#### Step 4: Example Calculation
Suppose we are given:
- One angle is \(30^\circ\).
- Another angle is \(150^\circ\).

We can verify:
- The angle \(30^\circ\) and \(150^\circ\) are supplementary because \(30^\circ + 150^\circ = 180^\circ\).
- The vertical angles opposite each other will be equal. So, if one angle is \(30^\circ\), its vertical angle will also be \(30^\circ\).
- Similarly, if one angle is \(150^\circ\), its vertical angle will also be \(150^\circ\).

#### Step 5: Generalize the Solution
For any intersecting lines:
- If one angle is \(x\), its vertical angle will also be \(x\).
- If one angle is \(y\), its vertical angle will also be \(y\).
- Adjacent angles will always sum to \(180^\circ\).

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Final Answer:

The key takeaway is that vertical angles are always equal. If you are given one angle, you can immediately determine its vertical angle. If you are given an adjacent angle, you can find its supplement.

\[
\boxed{\text{Vertical angles are always equal.}}
\]