Question

question 11 (1 point)
complete the two - column proof.
given: ∠1 and ∠3 are vertical angles.
prove: ∠1 ≅ ∠3
diagram of two intersecting lines forming angles 1, 2, 3
statements

1. ∠1 and ∠3 are vertical angles.
2. ∠1 and ∠2 are supplementary.

∠2 and ∠3 are supplementary.

1. ( mangle1 + mangle2 = 180 )

( mangle2 + mangle3 = 180 )

1. ( mangle1 + mangle2 = mangle2 + mangle3 )
2. ( mangle1 = mangle3 )
3. ( angle1 cong angle3 )

reasons

1. given
2. ∠s that form a linear pair are supplementary.

∠1 and ∠2 form a linear pair.
∠2 and ∠3 form a linear pair.

1. the sum of the measures of supplementary angles is 180.
2. ______
3. ______
4. ______

options:
○ a

1. addition property of equality
2. subtraction property of equality
3. ∠s with the same measure are ≅.

○ b

1. transitive property of equality
2. subtraction property of equality
3. ∠s with the same measure are ≅.

○ c

1. transitive property of equality
2. addition property of equality
3. ∠s with the same measure are ≅.

○ d

1. addition property of equality
2. transitive property of equality
3. ∠s with the same measure are ≅.

Explanation:

1. For step 4: We know \( m\angle1 + m\angle2 = 180 \) and \( m\angle2 + m\angle3 = 180 \). By the Transitive Property of Equality (if \( a = c \) and \( b = c \), then \( a = b \)), we can say \( m\angle1 + m\angle2 = m\angle2 + m\angle3 \).
2. For step 5: To get \( m\angle1 = m\angle3 \), we subtract \( m\angle2 \) from both sides of \( m\angle1 + m\angle2 = m\angle2 + m\angle3 \), which is the Subtraction Property of Equality.
3. For step 6: If two angles have the same measure (\( m\angle1 = m\angle3 \)), then the angles are congruent (\( \angle1\cong\angle3 \)) by the definition that angles with the same measure are congruent.

Looking at the options, option b has step 4 as Transitive Property of Equality, step 5 as Subtraction Property of Equality, and step 6 as \( \angle\text{s with the same measure are}\cong \).

Answer:

b. 4. Transitive Property of Equality; 5. Subtraction Property of Equality; 6. \( \angle\text{s with the same measure are}\cong \)