Prove the vertical angle theorem

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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 AB and line CD, intersecting at point O. The vertical angles formed are ∠AOC and ∠BOD, and the adjacent angles are ∠AOB and ∠COD.Use Angle Sum Property: Use the fact that the sum of angles around a point is 360 degrees. At point O, the sum of all angles around it is 360 degrees. Therefore, ∠AOC + ∠AOB + ∠BOD + ∠COD = 360 degrees.Express Adjacent Angles: Express adjacent angles as supplementary. Since ∠AOC and ∠AOB are adjacent, they are supplementary, which means ∠AOC + ∠AOB = 180 degrees. Similarly, ∠BOD and ∠COD are supplementary, so ∠BOD + ∠COD = 180 degrees.Substitute Supplementary Sums: Substitute the supplementary angle sums into the total sum around point O. Replace ∠AOC + ∠AOB with 180 degrees and ∠BOD + ∠COD with 180 degrees in the equation from step 4. This gives us 180 degrees + 180 degrees = 360 degrees, which is true.Conclude Angle Equality: Conclude that the vertical angles are equal. Since ∠AOC + ∠AOB = 180 degrees and ∠BOD + ∠COD = 180 degrees, and we know that ∠AOB = ∠COD (as they are also vertical angles), we can deduce that ∠AOC = ∠BOD.

Prove the vertical angle theorem

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