Question

Choose one of the following theorems and prove it: Vertical Angles are congruent, Alternate interior angles are congruent, Corresponding angles are congruent. Demonstrate on a whiteboard.

Answer 1

Given the question in the question tab, the following are the solution steps to prove the theorem

STEP 1: Write the theorem

Theorem: Vertical Angles are congruent

Step 2: Define Vertical Angles

When two lines intersect, four angles are formed. There are two pairs of nonadjacent angles. These pairs are called vertical angles. In the image given below, (∠1, ∠3) and (∠2, ∠4) are two vertical angle pairs.

STEP 3: State the Vertical angles theorem

Vertical angles theorem or vertically opposite angles theorem states that two opposite vertical angles formed when two lines intersect each other are always equal (congruent) to each other.

STEP 4: Prove the theorem

The proof is based on straight angles. We already know that angles on a straight line add up to 180°.

`\begin{gathered} <1+<2=180^(\circ)\text{ (linear pair of angles and equal to angle on a straight line)}--\text{equation 1} \\ <1+<4=180^(\circ)\text{ (linear pair of angles and equal to angle on a straight line)}---\text{equation }2 \\ \text{From equations (1) and (2),} \\ <1+<2=<1+<4=180^(\circ) \\ \text{According to transitive property, if a=b and b=c then a=c} \\ \text{Therefore, we can rewrite the statement as }<1+<2=<4---\text{equation 3} \\ By\text{ eliminating <1 on both sides of the equation (3), we get <2=<4} \\ \text{Similarly, we can use the same set of statements to prove that <1=<3.} \\ \text{Therefore, we conclude that vertically opposite angles are always equal.} \end{gathered}`