Question

Instruction: Proving Vertical Angles Are Congruent. Given: ∠22 and ∠24 are vertical angles. Prove: ∠22 ≅ ∠24.

Statements:

m∠22 + m∠23 = 180
m∠23 + m∠24 = 180
∠22 and ∠24 are vertical angles
∠22 and ∠23 are a linear pair
∠23 and ∠24 are a linear pair
Lines m and n intersect at P
Reasons:
A. 2
B. P
C. n
D. 4

Answer 1

To prove that vertical angles are congruent, we establish that the angles forming linear pairs with a common angle are supplementary. By the transitive property of equality, we then show that the vertical angles are equal in measure, thus proving congruence.

Step-by-step explanation:

To prove that vertical angles are congruent, we use the given statements and reasons to construct a logical argument. Given that ∠22 and ∠24 are vertical angles, we start with the two given equations m∠22 + m∠23 = 180 and m∠23 + m∠24 = 180. The reason behind these equations is that the angles form a linear pair with a third angle ∠23, making them supplementary (reason 'D' for both). Since these two separate sums equal 180 degrees, and both include m∠23, by the transitive property of equality (if a = b and b = c, then a = c), we deduce that m∠22 must be equal to m∠24. Thus, ∠22 and ∠24 are congruent due to the definition of congruent angles (reason 'C').