Question

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jamar draws three pairs of parallel lines that are each
intersected by a third line. in each figure, he measures a
pair of same - side interior angles.
what is a reasonable conjecture for jamar to make by
recognizing a pattern and using inductive reasoning?
when a pair of parallel lines are intersected by
a third line, the same - side interior angles are
acute.
when a pair of parallel lines are intersected by
a third line, the same - side interior angles are
supplementary.
when a pair of parallel lines are intersected by
a third line, all of the angles formed are
supplementary.

Explanation:

Same - side interior angles are angles that lie between two parallel lines and on the same side of a transversal. In the given figure, \(100^{\circ}+80^{\circ} = 180^{\circ}\). If we assume another pair of same - side interior angles (say \(x\) and \(y\) where \(x + y\) follows the same pattern as in the first - shown pair of parallel lines). The property of same - side interior angles when two parallel lines are cut by a transversal is that they are supplementary (i.e., their sum is \(180^{\circ}\)).
The first option is wrong because \(100^{\circ}\) is an obtuse angle. The third option is wrong because not all angles formed are supplementary. For example, adjacent angles (linear pairs) are supplementary, but non - adjacent angles (like vertical angles) are equal.

Answer:

When a pair of parallel lines are intersected by a third line, the same - side interior angles are supplementary.