Question

1. a || b and e || f given
2. blank corresponding angles theorem
3. m∠5 + m∠6 = 180° and m∠9 + m∠11 = 180° definition of linear pair
4. m∠13 + m∠6 = 180° and m∠13 + m∠11 = 180° blank
5. blank transitive property of equality
6. m∠6 = m∠11 blank

□ cpctc theorem □ m∠13 + m∠6 = m∠13 + m∠11
□ m∠5 = m∠13 and m∠9 = m∠13
□ m∠5 + m∠7 = 180° and m∠13 + m∠14 = 180° □ substitution property of equality
□ subtraction property of equality

Explanation:

Step1: Use corresponding - angles theorem

Since \(a\parallel b\) and \(e\parallel f\), by the Corresponding Angles Theorem, we have \(m\angle5 = m\angle13\) and \(m\angle9 = m\angle13\).

Step2: Apply linear - pair definition

We know that \(m\angle5 + m\angle6=180^{\circ}\) and \(m\angle9 + m\angle11 = 180^{\circ}\) from the definition of linear pair.

Step3: Substitute equal angles

Since \(m\angle5 = m\angle13\) and \(m\angle9 = m\angle13\), we can substitute to get \(m\angle13 + m\angle6 = 180^{\circ}\) and \(m\angle13 + m\angle11 = 180^{\circ}\).

Step4: Use transitive property

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

Step5: Use subtraction property

Subtract \(m\angle13\) from both sides of the equation \(m\angle13 + m\angle6=m\angle13 + m\angle11\) using the Subtraction Property of Equality to get \(m\angle6 = m\angle11\).

Answer:

1. \(m\angle5 = m\angle13\) and \(m\angle9 = m\angle13\)
2. Substitution Property of Equality
3. Subtraction Property of Equality