Question

statement reason
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} ) substitution 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 ) subtraction property of equality

Explanation:

Step1: Use Corresponding Angles Theorem

Since \(a\parallel b\), by Corresponding Angles Theorem, \(m\angle5 = m\angle13\). Since \(c\parallel f\), by Corresponding Angles Theorem, \(m\angle9 = m\angle13\).

Step2: Use Substitution Property of Equality

We know \(m\angle5 + m\angle6=180^{\circ}\) (linear - pair) and \(m\angle5 = m\angle13\), so substituting \(m\angle5\) with \(m\angle13\) gives \(m\angle13 + m\angle6 = 180^{\circ}\). Also, \(m\angle9 + m\angle11 = 180^{\circ}\) (linear - pair) and \(m\angle9 = m\angle13\), so substituting \(m\angle9\) with \(m\angle13\) gives \(m\angle13 + m\angle11=180^{\circ}\).

Step3: Use Transitive Property of Equality

Since \(m\angle13 + m\angle6 = 180^{\circ}\) and \(m\angle13 + m\angle11 = 180^{\circ}\), by Transitive Property of Equality \(m\angle13 + m\angle6=m\angle13 + m\angle11\).

Step4: Use Subtraction Property of Equality

Subtract \(m\angle13\) from both sides of the equation \(m\angle13 + m\angle6=m\angle13 + m\angle11\). We get \(m\angle6=m\angle11\) (because \(m\angle13 + m\angle6-(m\angle13)=m\angle13 + m\angle11-(m\angle13)\)).

Answer:

The missing statements and the step - by - step application of geometric theorems (Corresponding Angles Theorem, Substitution Property of Equality, Transitive Property of Equality, Subtraction Property of Equality) are used to prove \(m\angle6 = m\angle11\).