Question

complete the partially completed proof of the vertical angles theorem
given: ∠dqa and ∠aqb are a linear pair.
∠aqb and ∠bqc are a linear pair
prove: ∠dqa ≅ ∠bqc
complete the missing steps in the proof.

1. match a statement or reason to each space provided.

statements	reasons
2. ∠aqb and ∠bqc are a linear pair.	2. given
3. ∠dqa and ∠aqb are supplementary angles.	3. linear pair postulate
4. ∠aqb and ∠bqc are supplementary angles.	4. linear pair postulate
5. m∠dqa + m∠aqb = 180°	5. definition of supplementary angles.
6. m∠aqb + m∠bqc = 180°	6. definition of supplementary angles
7.	7.

statements	reasons
m∠dqa = m∠bqc	subtraction property of equality
m∠aqb = m∠aqb	reflexive property
∠dqa ≅ ∠bqc	definition of congruent angles

Explanation:

Step1: Use Substitution Property

From steps 5 and 6, we know \( m\angle DQA + m\angle AQB = 180^\circ \) and \( m\angle AQB + m\angle BQC = 180^\circ \). By the Substitution Property of Equality, we can set them equal: \( m\angle DQA + m\angle AQB = m\angle AQB + m\angle BQC \).

Step2: Subtract \( m\angle AQB \)

Subtract \( m\angle AQB \) from both sides of the equation \( m\angle DQA + m\angle AQB = m\angle AQB + m\angle BQC \). This gives \( m\angle DQA = m\angle BQC \).

Step3: Use Definition of Congruent Angles

By the definition of congruent angles (if the measures of two angles are equal, the angles are congruent), we conclude \( \angle DQA \cong \angle BQC \).

Answer:

For step 6 (Statement): \( m\angle DQA + m\angle AQB = m\angle AQB + m\angle BQC \) (using Substitution Property of Equality from steps 5 and 6)
Then \( m\angle DQA = m\angle BQC \) (subtracting \( m\angle AQB \) from both sides)
Finally, \( \angle DQA \cong \angle BQC \) (by definition of congruent angles)

(For the table, the last statement (step 6) would be \( \angle DQA \cong \angle BQC \) and the reason would be "Definition of congruent angles" after the previous algebraic steps.)