Question

29,622 Learners found this answer helpful

this is a contradiction.

1. conclude that the assumption is false and hence that the original conjecture must be true. the assumption that ∠1 and ∠2 form a linear pair is false. therefore ∠1 and ∠2 do not form a linear pair.

use the following statement for exercises 1–4.
an obtuse triangle cannot have a right angle.

1. identify the conjecture to be proven.

1. assume the opposite of the conclusion. write this assumption.

1. use direct reasoning to arrive at a contradiction.

1. what can you conclude?

Explanation:

1. Identify the conjecture to be proven.

The given statement is "An obtuse triangle cannot have a right angle." So the conjecture to prove is that an obtuse triangle does not contain a right angle.

The conclusion of the conjecture is that an obtuse triangle does not have a right angle. The opposite (negation) of this is that an obtuse triangle has a right angle.

Step 1: Recall the definitions. An obtuse triangle has one angle greater than \(90^\circ\) (let's call this angle \(\angle B>90^\circ\)), and a right triangle has one angle equal to \(90^\circ\) (let's say \(\angle C = 90^\circ\) if we assume the obtuse triangle has a right angle).
Step 2: The sum of the interior angles of a triangle is \(180^\circ\). So if a triangle has an obtuse angle (\(>90^\circ\)) and a right angle (\(=90^\circ\)), the sum of these two angles alone is \(>90^\circ + 90^\circ=180^\circ\), which contradicts the angle - sum property of a triangle (sum of interior angles of a triangle is \(180^\circ\)).

Answer:

An obtuse triangle cannot have a right angle (or: The conjecture is that an obtuse triangle does not have a right angle).

2. Assume the opposite of the conclusion. Write this assumption.