Question

for problems 4&5, use the diagram at the right

1. fill in the blanks with the following possible answers.

∠3
∠6
∠5
complementary
congruent
supplementary
linear pair
∠2
use the same-side interior angles postulate to show that ∠3 is supplementary to
show ∠6 and
are
because they form a
since ∠3 and ∠6 are
to the same angle, they are congruent
to each other.
lines ℓ and m are intersected by transversal t and ℓ ∥ m.

Explanation:

Step1: Analyze the relationship between angles

When two parallel lines \(l\) and \(m\) are cut by a transversal \(t\), we use angle - pair relationships.

Step2: Use the Same - Side Interior Angles Postulate

The Same - Side Interior Angles Postulate states that if two parallel lines are cut by a transversal, then the same - side interior angles are supplementary. Here, \(\angle3\) and \(\angle6\) are same - side interior angles.

Step3: Analyze the linear - pair relationship

\(\angle6\) and \(\angle5\) form a linear pair. By the definition of a linear pair, if two angles form a linear pair, then they are supplementary.

Step4: Use the congruent - angle property

Since \(\angle3\) and \(\angle6\) are supplementary (Same - Side Interior Angles Postulate) and \(\angle6\) and \(\angle5\) are supplementary (linear pair), then \(\angle3\cong\angle5\) (angles supplementary to the same angle are congruent)

Answer:

\(\angle3\) and \(\angle6\) are supplementary (Same - Side Interior Angles Postulate). \(\angle6\) and \(\angle5\) are supplementary (linear pair). \(\angle3\) and \(\angle5\) are congruent.