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Difficulty: MediumLines and Angles

In the figure, lines ABAB and CDCD intersect at point OO, and ray OEOE is perpendicular to line ABAB. If the measure of ∠EOC\angle EOC is 27∘27^\circ, what is the measure, in degrees, of ∠BOD\angle BOD?

Answer: 63

Answer

The measure of ∠BOD\angle BOD is 63 degrees.
The correct answer is 63. Since ray OEOE is perpendicular to line ABAB, the angle ∠EOA\angle EOA is a right angle measuring 90∘90^\circ. The adjacent angles ∠AOC\angle AOC and ∠EOC\angle EOC make up ∠EOA\angle EOA, which means ∠AOC=90∘−27∘=63∘\angle AOC = 90^\circ - 27^\circ = 63^\circ. Finally, because lines ABAB and CDCD intersect at point OO, the angle ∠BOD\angle BOD and the angle ∠AOC\angle AOC are vertical angles. Vertical angles are equal in measure, so the measure of ∠BOD\angle BOD is 63∘63^\circ.

Step-by-Step Solution

1
Identify the angle formed by perpendicular lines.
The measure of ∠EOA\angle EOA is 90∘90^\circ.
Since ray OEOE is perpendicular to line ABAB, the angle ∠EOA\angle EOA is a right angle.
2
Calculate the measure of ∠AOC\angle AOC.
The measure of ∠AOC\angle AOC is 63∘63^\circ.
Angles ∠AOC\angle AOC and ∠EOC\angle EOC are adjacent and form the right angle ∠EOA\angle EOA, meaning they are complementary: ∠AOC=90∘−27∘=63∘\angle AOC = 90^\circ - 27^\circ = 63^\circ.
3
Determine the measure of ∠BOD\angle BOD.
The measure of ∠BOD\angle BOD is 63∘63^\circ.
Lines ABAB and CDCD intersect at point OO, making ∠BOD\angle BOD and ∠AOC\angle AOC vertical angles. Since vertical angles are equal, the measure of ∠BOD\angle BOD is equal to the measure of ∠AOC\angle AOC.

Key Concept

Using properties of perpendicular lines and vertical angles to solve for unknown angle measures.
Estimated Time:1m 30s
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Topics

Lines and Angles

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