Question

Complete the following indirect proof (proof by contradiction).

Given: Adjacent angles LA and ZB, formed by the intersection of two lines
Prove: At least one of the angles LA and B has measure 90° or greater

Answer 1

First, we assume that this conclusion is false. In other words, we assume that the contrary statement "none of the two angles has measure
`90^(\circ)` or greater" is true.

The assumption is equivalent to the following two statements:

(1)
`m\angle A\text{ } \boxed{ < 90^(\circ)}`

(2)
`m\angle B\text{ } \boxed{ < 90^(\circ)}`

Using (1) and (2) and the addition properties of inequalities, we conclude that
`m\angle A+m\angle B \text{ } \boxed{ < } \text{ } 180^(\circ)`.

On the other hand, two adjacent angles form a linear pair. Thus, the last statement contradicts the Linear Pair Property, which states that for a linear pair of angles
`\angle A` and
`\angle B`,
`m\angle A+m\angle B \text{ } \boxed{=} \text{ } 180^(\circ)`.

Therefore, the assumption made is false, and the statement "at least one of the angles
`\angle A` and
`\angle B` has measure
`90^(\circ)` or greater" is true.