Question

statements

1. ∠5 and ∠7 are vertical angles.
2. ∠5 and ∠6 are a linear pair. ∠6 and ∠7 are a linear pair.
3. ∠5 and ∠6 are supplementary. ∠6 and ∠7 are supplementary.
4. ∠5 ≅ ∠7

reasons

1. given
2. definition of linear pair, as shown in the diagram
3. linear pair postulate
4. congruent supplements theorem

rewrite the two - column proof without using the congruent supplements theorem.

statements
1.

1. ∠5 and ∠6 are a linear pair. ∠6 and ∠7 are a linear pair.
2. ∠5 and ∠6 are supplementary. ∠6 and ∠7 are supplementary.
3. and
4. ( mangle5 + mangle6 = mangle6 + mangle7 )

6.
7.
reasons

1. given

2.
3.

1. definition of supplementary angles

5.

1. subtraction property of equality

7.

Explanation:

Step1: Fill Statement 1

$\angle 5$ and $\angle 7$ are vertical angles. (Given)

Step2: Fill Reason 2

Definition of linear pair, as shown in the diagram.

Step3: Fill Reason 3

Linear Pair Postulate.

Step4: Fill Statement 4

$m\angle 5 + m\angle 6 = 180^\circ$ and $m\angle 6 + m\angle 7 = 180^\circ$ (By definition of supplementary angles, since $\angle 5$ & $\angle 6$, $\angle 6$ & $\angle 7$ are supplementary)

Step5: Fill Reason 5

Substitution Property (Substitute the equal sums from Step 4)

Step6: Fill Statement 6

$m\angle 5 = m\angle 7$ (Subtract $m\angle 6$ from both sides using Subtraction Property of Equality)

Step7: Fill Reason 7

Definition of congruent angles (If $m\angle 5 = m\angle 7$, then $\angle 5 \cong \angle 7$)

Answer:

1. $\boldsymbol{\angle 5}$ and $\boldsymbol{\angle 7}$ are vertical angles.
2. Definition of linear pair, as shown in the diagram.
3. Linear Pair Postulate.
4. $\boldsymbol{m\angle 5 + m\angle 6 = 180^\circ}$ and $\boldsymbol{m\angle 6 + m\angle 7 = 180^\circ}$
5. Substitution Property
6. $\boldsymbol{m\angle 5 = m\angle 7}$
7. Definition of congruent angles