Prove the Vertical Pair Theorem: If two angles form a vertical pair, then they have the same measure.
The Correct Answer and Explanation is :
The Vertical Pair Theorem states that if two angles form a vertical pair, they have the same measure. Vertical angles are formed when two lines intersect, and the angles opposite each other at the intersection are vertical angles.
Proof of the Vertical Pair Theorem:
Consider two intersecting lines forming four angles. Label the angles as ∠1, ∠2, ∠3, and ∠4. By definition, vertical angles are pairs of opposite angles formed by two intersecting lines.
- Identify Vertical Angles: Suppose ∠1 and ∠2 are adjacent, and ∠1 and ∠3 are vertical angles. Similarly, ∠2 and ∠4 are adjacent, and ∠3 and ∠4 are vertical angles.
- Use the Linear Pair Property: The adjacent angles ∠1 and ∠2 form a linear pair, so they must add up to 180°. This can be written as:
[
\text{∠1} + \text{∠2} = 180^\circ
]
Similarly, ∠3 and ∠4 form a linear pair, so:
[
\text{∠3} + \text{∠4} = 180^\circ
] - Substitute Values: Now, because ∠1 and ∠3 are vertical angles, ∠1 = ∠3. Similarly, ∠2 = ∠4. Therefore, we can write:
[
\text{∠1} + \text{∠2} = 180^\circ
]
[
\text{∠3} + \text{∠4} = 180^\circ
]
Since ∠1 = ∠3, we substitute ∠3 for ∠1 in the second equation:
[
\text{∠1} + \text{∠4} = 180^\circ
] - Conclusion: From this, we can conclude that ∠1 = ∠4, proving that vertical angles are congruent. Therefore, vertical angles have the same measure.
Explanation:
Vertical angles are formed when two lines intersect. The theorem proves that the angles opposite each other at the intersection are congruent (have the same measure). The proof relies on the concept of linear pairs, where adjacent angles on a straight line sum to 180°. By setting up equations and substituting congruent angles, we show that vertical angles must have equal measures.
Let me also create an image to illustrate this proof.
Here is the illustration of the two intersecting lines forming four angles. As shown in the image, the opposite angles (Angle 1 and Angle 3, Angle 2 and Angle 4) are vertical angles and are congruent, confirming the Vertical Pair Theorem.