Vertical Angles -Theorem, Proof, Vertically Opposite Angles - Free Printable

Problem Analysis:

The image shows a geometric figure with several intersecting lines and angles. The task is to determine the measure of the angle labeled as \( f \). Here's a step-by-step explanation of how to solve this problem:

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Step 1: Identify Key Information

- The point \( O \) is the center of intersection for multiple lines.
- There is a right angle marked between lines \( OB \) and \( OC \), indicating that \( \angle BOC = 90^\circ \).
- The angle \( \angle DOE \) is given as \( 118^\circ \).
- We need to find the measure of the angle \( f \), which is part of the configuration around point \( O \).

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Step 2: Use the Properties of Angles Around a Point

The sum of all angles around a point is always \( 360^\circ \). Therefore, we can write:
\[
\angle AOB + \angle BOC + \angle COD + \angle DOE = 360^\circ
\]

From the diagram:
- \( \angle BOC = 90^\circ \) (right angle).
- \( \angle DOE = 118^\circ \) (given).

We need to determine \( \angle AOB \) and \( \angle COD \) to find \( f \).

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Step 3: Express \( \angle AOB \) and \( \angle COD \)

Let:
- \( \angle AOB = x \)
- \( \angle COD = y \)

Using the total angle sum around point \( O \):
\[
x + 90^\circ + y + 118^\circ = 360^\circ
\]

Simplify:
\[
x + y + 208^\circ = 360^\circ
\]
\[
x + y = 152^\circ
\]

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Step 4: Determine \( f \)

The angle \( f \) is part of the configuration involving \( \angle BOC \) and \( \angle COD \). Specifically, \( f \) is the angle between \( OB \) and the line extending from \( O \) through the arc labeled \( f \). This arc is part of the sector formed by subtracting \( \angle BOC \) from \( \angle COD \).

Since \( \angle BOC = 90^\circ \) and \( \angle COD = y \), the angle \( f \) is:
\[
f = y - 90^\circ
\]

From the earlier equation \( x + y = 152^\circ \), we can express \( y \) as:
\[
y = 152^\circ - x
\]

However, we do not need the exact value of \( x \) to find \( f \). Instead, we use the fact that \( f \) is directly related to \( y \) and the right angle \( 90^\circ \).

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Step 5: Solve for \( f \)

Since \( f = y - 90^\circ \) and \( y = 152^\circ - x \), we can substitute:
\[
f = (152^\circ - x) - 90^\circ
\]
\[
f = 62^\circ
\]

Thus, the measure of angle \( f \) is:
\[
\boxed{62}
\]