Question

complete the table to prove congruency between angles
statement\treason

1. a || b and c || f\tgiven
2. m∠5 = m∠13 and m∠9 = m∠13\tcorresponding angles theorem
3. m∠5 + m∠6 = 180° and m∠9 + m∠11 = 180°\tdefinition of linear pair
4. m∠13 + m∠6 = 180° and m∠13 + m∠11 = 180°\tsubstitution property of equality
5. m∠5 + m∠7 = 180° and m∠13 + m∠14 = 180°\ttransitive property of equality
6. m∠6 = m∠11\tsubtraction property of equality

Explanation:

Step1: Given parallel - lines

We are given \(a\parallel b\) and \(c\parallel f\).

Step2: Apply corresponding - angles theorem

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

Step3: Use linear - pair definition

By the definition of a linear pair, \(m\angle5 + m\angle6=180^{\circ}\) and \(m\angle9 + m\angle11 = 180^{\circ}\).

Step4: Substitute equal angles

Because \(m\angle5 = m\angle13\) and \(m\angle9 = m\angle13\), we substitute in the linear - pair equations to get \(m\angle13 + m\angle6=180^{\circ}\) and \(m\angle13 + m\angle11 = 180^{\circ}\) using the Substitution Property of Equality.

Step5: Analyze transitive property

We have \(m\angle5 + m\angle6=180^{\circ}\), \(m\angle13 + m\angle6=180^{\circ}\), \(m\angle9 + m\angle11 = 180^{\circ}\), \(m\angle13 + m\angle11 = 180^{\circ}\). From \(m\angle5 + m\angle6=180^{\circ}\) and \(m\angle13 + m\angle6=180^{\circ}\), and \(m\angle9 + m\angle11 = 180^{\circ}\), \(m\angle13 + m\angle11 = 180^{\circ}\), by the Transitive Property of Equality, we can get other linear - pair relationships (but the step \(m\angle5 + m\angle7 = 180^{\circ}\) and \(m\angle13 + m\angle14 = 180^{\circ}\) seems to be an error or non - relevant to the final proof of \(m\angle6=m\angle11\)).

Step6: Prove angle equality

From \(m\angle13 + m\angle6=180^{\circ}\) and \(m\angle13 + m\angle11 = 180^{\circ}\), we subtract \(m\angle13\) from both equations.
\(m\angle13 + m\angle6-(m\angle13)=180^{\circ}-m\angle13\) and \(m\angle13 + m\angle11-(m\angle13)=180^{\circ}-m\angle13\)
By the Subtraction Property of Equality, we get \(m\angle6 = m\angle11\).

Answer:

The steps in the table are correctly justified as shown above to prove \(m\angle6 = m\angle11\).