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Prove the vertical angle theorem
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Q. Prove the vertical angle theorem
- Identify Vertical Angles: Identify the vertical angles in a pair of intersecting lines.When two lines intersect, they form two pairs of opposite, or vertical, angles. These angles are called vertical angles.
- State Theorem: State the Vertical Angle Theorem.The Vertical Angle Theorem states that the vertical angles formed by two intersecting lines are congruent, meaning they have equal measures.
- Set Up Proof: Set up a proof using arbitrary angles.Let's denote the intersecting lines as line and line , intersecting at point . The vertical angles formed are and , and the adjacent angles are and .
- Use Angle Sum Property: Use the fact that the sum of angles around a point is degrees.At point , the sum of all angles around it is degrees. Therefore, degrees.
- Express Adjacent Angles: Express adjacent angles as supplementary. Since and are adjacent, they are supplementary, which means degrees. Similarly, and are supplementary, so degrees.
- Substitute Supplementary Sums: Substitute the supplementary angle sums into the total sum around point O.Replace with degrees and with degrees in the equation from step . This gives us degrees degrees degrees, which is true.
- Conclude Angle Equality: Conclude that the vertical angles are equal.Since degrees and degrees, and we know that (as they are also vertical angles), we can deduce that .
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