Question

complete the table to prove congruency between angles

1. ( a parallel b ) and ( c parallel f ) given
2. ( m angle 5 = m angle 13 ) and ( m angle 9 = m angle 13 ) corresponding angles theorem
3. ( m angle 5 + m angle 6 = 180^{circ} ) and ( m angle 9 + m angle 11 = 180^{circ} ) definition of linear pair
4. ( m angle 13 + m angle 6 = 180^{circ} ) and ( m angle 13 + m angle 11 = 180^{circ} ) subtraction property of equality
5. ( m angle 13 + m angle 6 = m angle 13 + m angle 11 ) transitive property of equality
6. ( m angle 6 = m angle 11 ) cpctc theorem

Explanation:

Step1: Use Corresponding Angles Theorem

Since \(a\parallel b\) and \(c\parallel f\), by the Corresponding Angles Theorem, if two parallel lines are cut by a transversal, corresponding angles are congruent. So \(m\angle5 = m\angle13\) (transversal \(c\) cuts \(a\) and \(b\)) and \(m\angle9 = m\angle13\) (transversal \(f\) cuts \(a\) and \(b\)).

Step2: Substitute using linear - pair equations

We know \(m\angle5 + m\angle6=180^{\circ}\) and \(m\angle9 + m\angle11 = 180^{\circ}\). Since \(m\angle5 = m\angle13\) and \(m\angle9 = m\angle13\), we substitute \(m\angle5\) with \(m\angle13\) in \(m\angle5 + m\angle6 = 180^{\circ}\) to get \(m\angle13 + m\angle6=180^{\circ}\), and substitute \(m\angle9\) with \(m\angle13\) in \(m\angle9 + m\angle11 = 180^{\circ}\) to get \(m\angle13 + m\angle11 = 180^{\circ}\). The reason for this substitution is the Substitution Property of Equality.

Step3: Use Transitive Property

Since \(m\angle13 + m\angle6=180^{\circ}\) and \(m\angle13 + m\angle11 = 180^{\circ}\), by the Transitive Property of Equality (\(a = b\) and \(a = c\) implies \(b = c\)), we have \(m\angle13 + m\angle6=m\angle13 + m\angle11\).

Step4: Use Subtraction Property

Subtract \(m\angle13\) from both sides of the equation \(m\angle13 + m\angle6=m\angle13 + m\angle11\). By the Subtraction Property of Equality (\(a + b=a + c\) implies \(b = c\)), we get \(m\angle6=m\angle11\).

Answer:

The missing statements and reasons:

• Statement 2: \(m\angle5 = m\angle13\) (because \(a\parallel b\) and \(c\) is a transversal, Corresponding Angles Theorem) and \(m\angle9 = m\angle13\) (because \(a\parallel b\) and \(f\) is a transversal, Corresponding Angles Theorem)
• Reason 4: Substitution Property of Equality
• Reason 6: Subtraction Property of Equality