Proof that the sum of a linear pair of angles is 180 degrees

A linear pair of angles consists of two adjacent angles formed when two lines intersect. To prove that the sum of a linear pair of angles is 180 degrees, we will use the following steps:

Step 1: Understanding a Linear Pair of Angles
A linear pair of angles is formed when two adjacent angles are created by the intersection of two lines. The angles share a common vertex and a common side while their non-common sides are opposite rays.

Step 2: Labeling the Linear Pair of Angles
Let's consider a linear pair of angles and label them as ∠A and ∠B. Here, ∠A and ∠B are adjacent angles formed by the lines AB and CD, intersecting at point P.

Step 3: Defining a Straight Angle
A straight angle is formed when two opposite rays create a straight line. A straight angle measures 180 degrees.

Step 4: Proving the Sum of a Linear Pair of Angles
To prove that the sum of a linear pair of angles is 180 degrees, we need to show that ∠A + ∠B = 180 degrees.

- Draw a line segment joining point A to point B.
- Extend the line segment AB to form a straight angle with the line segment CD.
- This creates a straight line, which we'll label as line EF.
- As per step 3, a straight angle measures 180 degrees.
- Now, we can observe that ∠A + ∠B + ∠E = 180 degrees, as the angles ∠A, ∠B, and ∠E collectively form a straight angle.
- However, ∠E is equal to 0 degrees since it is a straight angle formed by the extension of the line segment AB.
- Therefore, ∠A + ∠B + 0 degrees = 180 degrees.
- Simplifying the equation, we have ∠A + ∠B = 180 degrees.

Step 5: Conclusion
From the above steps, we can conclude that the sum of a linear pair of angles (∠A and ∠B) is indeed 180 degrees.

A liner pair of angles is formed when two lines intersect.