Question

given: ∠1 and ∠2 are vertical angles
prove: ∠1 ≅ ∠2

1. ∠1 and ∠2 are vertical angles 1. given
2. ∠1 and ∠3 form a linear pair 2. definition of linear pair

∠2 and ∠3 form a linear pair

1. ∠1 and ∠2 are supplementary 3. linear pairs are supplementary

∠2 and ∠3 are supplementary

1. m∠1 + m∠3 = 180 4. definition of supplementary

m∠2 + m∠3 = 180

1. m∠1 + m∠3 = m∠2 + m∠3 5.
2. m∠3 = m∠3 6.
3. m∠1 = m∠2 7.
4. ∠1 ≅ ∠2 8.

substitution reflexive property
subtraction poe def. of congruent

Explanation:

Step1: Analyze statement 5

Since \(m\angle1 + m\angle3=180\) and \(m\angle2 + m\angle3 = 180\), we can substitute the second equation into the first. So the reason for statement 5 is substitution.

Step2: Analyze statement 6

The reflexive property states that a quantity is equal to itself. So \(m\angle3=m\angle3\) is by the reflexive property.

Step3: Analyze statement 7

If \(m\angle1 + m\angle3=m\angle2 + m\angle3\), we can subtract \(m\angle3\) from both sides. By the subtraction property of equality (subtraction PoE), \(m\angle1=m\angle2\).

Step4: Analyze statement 8

If \(m\angle1 = m\angle2\), then by the definition of congruent angles (def. of congruent), \(\angle1\cong\angle2\).

Answer:

1. substitution; 6. reflexive property; 7. subtraction PoE; 8. def. of congruent