Q:

Kathryn draws three pairs of intersecting lines. In each figure, she measures a pair of angles. What is a reasonable conjecture for Kathryn to make by recognizing a pattern and using inductive reasoning? When a pair of lines intersect, the vertical angles are acute. When a pair of lines intersect, all of the angles formed are right angles. When a pair of lines intersect, the vertical angles are congruent. When a pair of lines intersect, all of the angles formed are congruent.

Accepted Solution

A:
Inductive reasoning is that when two lines intersect you will always have a pair of congruent angles considering an exception when all the angles are congruent. According to this,
1) Vertical angles (or opposite angles) are congruent. In this case, vertical angles could be acute or obtuse depending on the position of the lines.

2) The second case is the exception when all the angles are equal. It means that the lines are perpendicular.

3) Again back to the first case, vertical angles are equal or congruent in other words.

4) And again back to the second case when the lines are perpendicular, then all the angles that formed are equal to each other.

You can observe all the cases in the picture attached below.