Question

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 \(a\parallel b\) and \(e\parallel f\), by Corresponding Angles Theorem, \(m\angle5 = m\angle13\) (because \(\angle5\) and \(\angle13\) are corresponding angles) and \(m\angle9 = m\angle13\) (because \(\angle9\) and \(\angle13\) are corresponding angles).

Step2: Use Substitution Property of Equality

We know that \(m\angle5 + m\angle6=180^{\circ}\) (definition of linear pair) and \(m\angle9 + m\angle11 = 180^{\circ}\) (definition of linear pair). Substituting \(m\angle5\) with \(m\angle13\) (from \(m\angle5 = m\angle13\)) in \(m\angle5 + m\angle6=180^{\circ}\), we get \(m\angle13 + m\angle6 = 180^{\circ}\). Substituting \(m\angle9\) with \(m\angle13\) (from \(m\angle9 = m\angle13\)) in \(m\angle9 + m\angle11 = 180^{\circ}\), we get \(m\angle13 + m\angle11 = 180^{\circ}\).

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 (\(a = c\) and \(b = c\) implies \(a = b\)), we have \(m\angle13 + m\angle6=m\angle13 + m\angle11\).

Step4: Use Subtraction Property of Equality

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

Answer:

The proof shows that \(m\angle6 = m\angle11\) by using properties of parallel lines (Corresponding Angles Theorem), properties of equality (Substitution, Transitive, and Subtraction Properties) and the definition of a linear pair.