Understanding Angles Formed by Intersecting Lines

When two lines intersect at a point, they form four angles around that point. These angles have specific relationships, such as forming linear pairs and being vertically opposite to each other. This problem involves finding the measures of these four angles given a relationship between two adjacent angles.

Key Angle Concepts

To solve this problem, we need to recall two important geometric concepts:

• Linear Pair: Two adjacent angles that form a straight line. The sum of angles in a linear pair is always \(180^\circ\).
• Vertically Opposite Angles: When two lines intersect, the angles opposite each other at the intersection point are called vertically opposite angles. Vertically opposite angles are always equal in measure.

Solving the Problem: Finding the Angles

We are given two lines AB and CD that intersect at point O. The angles formed at O are \(\angle AOC\), \(\angle AOD\), \(\angle BOD\), and \(\angle BOC\).

We are given the relationship: \(\angle AOC = 5 \angle AOD\).

Let's denote \(\angle AOD\) as \(x\). According to the given information, \(\angle AOC = 5x\).

Using the Linear Pair Property

Angles \(\angle AOC\) and \(\angle AOD\) form a linear pair on the line CD (or AB). Therefore, their sum is \(180^\circ\).

\(\angle AOC + \angle AOD = 180^\circ\)

Substitute the expressions in terms of \(x\):

\(5x + x = 180^\circ\)

Combine the terms:

\(6x = 180^\circ\)

Solve for \(x\):

\(x = \frac{180^\circ}{6}\)

\(x = 30^\circ\)

Calculating the Measures of Angles

Now that we have the value of \(x\), we can find the measures of \(\angle AOD\) and \(\angle AOC\):

• \(\angle AOD = x = 30^\circ\)
• \(\angle AOC = 5x = 5 \times 30^\circ = 150^\circ\)

Using the Vertically Opposite Angles Property

Now we can find the measures of the other two angles using the property of vertically opposite angles:

• \(\angle BOD\) is vertically opposite to \(\angle AOC\). Therefore, \(\angle BOD = \angle AOC = 150^\circ\).
• \(\angle BOC\) is vertically opposite to \(\angle AOD\). Therefore, \(\angle BOC = \angle AOD = 30^\circ\).

Summary of the Four Angles

The four angles formed at the intersection point O are:

• \(\angle AOC = 150^\circ\)
• \(\angle AOD = 30^\circ\)
• \(\angle BOD = 150^\circ\)
• \(\angle BOC = 30^\circ\)

Listing the four angles in numerical order gives: \(30^\circ, 30^\circ, 150^\circ, 150^\circ\).

These angles \(30^\circ, 30^\circ, 150^\circ, 150^\circ\) correspond to one of the given options.

Revision Table: Key Geometric Properties

Additional Information on Intersecting Lines and Angles

When lines intersect, they create pairs of angles with specific properties. Understanding these properties is crucial for solving geometry problems involving intersecting lines. Apart from linear pairs and vertically opposite angles, other angle relationships exist when parallel lines are involved, such as corresponding angles, alternate interior angles, and consecutive interior angles. However, for simple intersecting lines without parallelism, linear pairs and vertically opposite angles are the primary concepts.

Remember that the sum of all four angles around the intersection point O must be \(360^\circ\). Let's check our result: \(30^\circ + 150^\circ + 30^\circ + 150^\circ = 360^\circ\). This confirms our calculations are correct.