Question

If two angles form a pair of vertical angles, then they are congruent

given ∠1 and ∠3 are vertical angles.
Prove ∠1≅∠3
You are given that ∠1 and ∠3 are verrtical angles.

Answers for the blanks:
A. supplementary angles _________
B. substitution ___________
C. addition ___________
D. subtraction ___________
E.symmetric ___________
F. m∠1≅m∠1 ___________
G. ∠1 = ∠3 ___________

Answer 1

To prove that vertical angles ∠1 and ∠3 are congruent, one must show that both pairs of angles (∠1 and ∠2, ∠3 and ∠2) are supplementary, and that through subtraction, their measures are equal, confirming ∠1 ≅ ∠3.

Step-by-step explanation:

If two angles form a pair of vertical angles, then they are congruent. This can be shown through a series of logical steps, making use of geometric principles. Given that ∠1 and ∠3 are vertical angles, we can proceed with the following proof:

1. ∠1 and ∠2 are supplementary angles because they are adjacent and their common side is a line. This means that m∠1 + m∠2 = 180°. (A. supplementary angles)
2. Similarly, ∠3 and ∠2 are supplementary angles, so m∠3 + m∠2 = 180°. (A. supplementary angles)
3. By the properties of equality, if m∠1 + m∠2 = m∠3 + m∠2, we can use subtraction to remove m∠2 from both equations. This gives us m∠1 = m∠3. (D. subtraction)
4. Using the transitive property of equality, if m∠1 = m∠3, then ∠1 is congruent to ∠3. (E. symmetric)

This proof demonstrates that vertical angles are congruent, which is shown by the equal measures m∠1 ≅ m∠3.