Question

Jamar draws three pairs of parallel lines that are each intersected by a third line. In each figure, he measures a pair of 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.

When a pair of parallel lines are intersected by a third line, all of the angles formed are acute.

Answer 1

Let us observe the figure, here a pair of parallel lines are intersected by a third line.

Here we can observe that angle 1 and 2 forms a linear pair.So, $\angle 1 + \angle 2 = 180^\circ$

Angle 1, 3 forms a linear pair. So, $\angle 1 + \angle 3 = 180^\circ$

Angle 2, 4 forms a linear pair. So, $\angle 2 + \angle 4 = 180^\circ$

Angle 3, 4 forms a linear pair. So, $\angle 3 + \angle 4 = 180^\circ$

Angle 3, 5 are interior angles on the same side of the transversal. So, these angles are supplementary. So, $\angle 3 + \angle 5 = 180^\circ$.

Angle 4, 6 are interior angles on the same side of the transversal. So, these angles are supplementary. So, $\angle 4 + \angle 6 = 180^\circ$.

Angle 5, 6 forms a linear pair. So, $\angle 5 + \angle 6 = 180^\circ$.

Angle 5, 7 forms a linear pair. So, $\angle 5 + \angle 7 = 180^\circ$.

Angle 6, 8 forms a linear pair. So, $\angle 6+ \angle 8 = 180^\circ$.

Angle 7, 8 forms a linear pair. So, $\angle 7 + \angle 8 = 180^\circ$.

So, All of the angles formed are supplementary.

Therefore, When a pair of parallel lines are intersected by a third line, all of the angles formed are supplementary.

So, Option C is the correct answer.

Answer 2

1 is not true generally, while 3 and 4 are never true. 2 is the only generally true conjecture.