Question

select the correct answer from each drop - down menu.
in the figure, ( a parallel b ), and both lines are intersected by transversal ( t ). complete the statements to prove that ( mangle1 = mangle5 ).
( a parallel b ) (given)
( mangle1 + mangle3 = 180^{circ} ) (linear pair theorem)
( mangle5 + mangle6 = 180^{circ} ) (linear pair theorem)
( mangle1 + mangle3=angle5+angle6 ) (
( mangle3 = mangle6 ) (
( mangle1 = mangle5 ) (subtraction p

Explanation:

Step1: Substitute equal values

Since \(m\angle1 + m\angle3=180^{\circ}\) and \(m\angle5 + m\angle6 = 180^{\circ}\), by the Transitive Property of Equality (if \(a = c\) and \(b=c\), then \(a = b\)), we have \(m\angle1 + m\angle3=m\angle5 + m\angle6\).

Step2: Use angle - relationship theorem

\(\angle3\) and \(\angle6\) are alternate interior angles. When \(a\parallel b\) (given) and \(t\) is a transversal, by the Alternate Interior Angles Theorem (if two parallel lines are cut by a transversal, then alternate interior angles are congruent), \(m\angle3=m\angle6\).

Answer:

First blank: Transitive Property of Equality; Second blank: Alternate Interior Angles Theorem.