Question

in this diagram, ( a parallel b ) and ( e parallel f ). sylvia writes a proof to prove that opposite angles, ( angle 6 ) and ( angle 11 ), are congruent in the parallelogram. drag and drop the statements and reasons into their correct locations in the two - column proof. (2 points)
complete the table to prove congruency between angles

statement	reason
2.	corresponding angles theorem
3. ( mangle 5 + mangle 6 = 180^{circ} ) and ( mangle 9 + mangle 11 = 180^{circ} )	definition of linear pair
4. ( mangle 12 + mangle 6 = 180^{circ} ) and ( mangle 13 + mangle 11 = 180^{circ} )
5.	transitive property of equality
6. ( mangle 6 = mangle 11 )

Explanation:

Step1: Use Corresponding Angles Theorem

Since \(a\parallel b\) and \(e\parallel f\), by the Corresponding Angles Theorem, \(\angle5=\angle9\) and \(\angle13 = \angle1\). But for the second - row (statement), we know that when two parallel lines are cut by a transversal, corresponding angles are congruent. So the statement for row 2 is \(\angle6=\angle14\) (because line \(e\parallel f\) and line \(a\) is the transversal, corresponding angles are congruent)

Step2: Use Definition of Linear Pair

For row 4, since \(a\parallel b\) and \(f\) is a transversal, \(\angle13+\angle6 = 180^{\circ}\) (linear - pair of angles formed by transversal \(f\) intersecting parallel lines \(a\) and \(b\)) and \(\angle13+\angle11=180^{\circ}\) (linear - pair of angles formed by transversal \(f\) intersecting parallel lines \(e\) and \(f\))

Step3: Use Transitive Property of Equality

From \(m\angle5 + m\angle6=180^{\circ}\), \(m\angle9 + m\angle11 = 180^{\circ}\), \(m\angle13 + m\angle6=180^{\circ}\), \(m\angle13 + m\angle11=180^{\circ}\), by the transitive property of equality \(m\angle5 + m\angle6=m\angle13 + m\angle6\) (because \(m\angle5=m\angle13\) from corresponding angles when \(a\parallel b\) and \(e\) is the transversal) and \(m\angle9 + m\angle11=m\angle13 + m\angle11\)

Step4: Use Subtraction Property of Equality

If \(m\angle5 + m\angle6=m\angle13 + m\angle6\) and \(m\angle9 + m\angle11=m\angle13 + m\angle11\), then subtracting \(m\angle6\) from the first equation and \(m\angle11\) from the second equation (Subtraction Property of Equality), we get \(m\angle5=m\angle13\) and \(m\angle9=m\angle13\), so \(m\angle6=m\angle11\)

Answer:

1. \(\angle6=\angle14\)
2. Definition of linear - pair (for parallel lines \(a\parallel b\) and transversal \(f\))
3. \(m\angle5 + m\angle6=m\angle13 + m\angle6\) and \(m\angle9 + m\angle11=m\angle13 + m\angle11\)
4. Subtraction Property of Equality