Question

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.

Answer 1

The reasonable conjecture for Kathryn to make by recognizing a pattern and using inductive reasoning is:

When a pair of lines intersect, the vertical angles are congruent.

Explanation:

The vertical angles are the opposite angles that are formed by the intersection of two lines. They are also known as Vertically opposite angles.

We know that when a pair of lines intersect then the vertical angles are always congruent.

However the measure of the angles may be acute or obtuse or right angle.

The same could be seen with the help of the figure attached to the answer.

Answer 2

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.