lines intersect at point A. One line contains points C, A, E and the other line contains points B, A, D. Which are linear pairs? Check all that apply. ∠DAE and ∠EAD ∠BAC and ∠CAD ∠BAE and ∠EAD ∠EAD and ∠DAC ∠CAE and ∠BAD
Correct Answer
- ∠BAC and ∠CAD
- ∠BAE and ∠EAD
- ∠EAD and ∠DAC
Explanation
In geometry, a linear pair is a set of two angles that satisfy two specific conditions:
- The angles must be adjacent, meaning they share a common vertex and a common side but do not have any interior points in common.
- Their non-common sides must be opposite rays, meaning they form a straight line.
A key consequence of this definition is that the measures of the angles in a linear pair always add up to 180 degrees.
Let’s analyze the given geometric setup. We have two lines, one containing points C, A, and E, and the other containing points B, A, and D. They intersect at point A. This means that line CAE and line BAD are straight lines.
We can now evaluate each pair of angles:
- ∠BAC and ∠CAD: These angles are adjacent because they share vertex A and the common side AC. Their non-common sides are ray AB and ray AD. Since points B, A, and D lie on a straight line, ray AB and ray AD are opposite rays. Thus, ∠BAC and ∠CAD are a linear pair.
- ∠BAE and ∠EAD: These angles are adjacent as they share vertex A and the common side AE. Their non-common sides are ray AB and ray AD, which we already identified as opposite rays forming the straight line BD. Therefore, ∠BAE and ∠EAD are a linear pair.
- ∠EAD and ∠DAC: These angles are adjacent, sharing vertex A and the common side AD. Their non-common sides are ray AE and ray AC. Since points C, A, and E lie on a straight line, ray AE and ray AC are opposite rays. Consequently, ∠EAD and ∠DAC are a linear pair.
- ∠DAE and ∠EAD: This is not a pair of two distinct angles. ∠DAE and ∠EAD represent the exact same angle, simply named with the points in a different order. An angle cannot form a linear pair with itself.
- ∠CAE and ∠BAD: These are not adjacent angles. While they share vertex A, they do not share a common side. In fact, both are straight angles formed by the intersecting lines. They cannot form a linear pair.
