An angle bisector always creates two acute angles. find a counterexample to show that the conjecture is false.

An acute angle is an angle that is less than 90°. An angle bisector is a ray drawn along an angle that bisects it into two equal and adjacent parts. Now, if the total angle is, say 270°, which is more than a half circle, it would result to two 135-degree angles. In this case, the angle is no longer acute, but obtuse.

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akposevictor akposevictor · Answer 2 · 2021-10-04T14:26:37+02:00

A counterexample showing that an angle bisector will not always create two acute angles is when an angle bisector bisects angle 200 degrees. (see figure 2 in the attachment).

Recall the following about acute angles, obtuse angles, and angle bisector:

The size of an acute angle is less than 90 degrees. (See Figure 1. Acute angle is labelled angle a).

The size of an obtuse angle is greater than 90 degrees. (See Figure 1. Obtuse angle is labelled angle b).

Angle bisector divides an angle into two equal halves as shown in figure 2. However, the angle divided is 200 degrees.

When an angle bisector divides an angle like 200 degrees, the resultant angles are above 90 degrees, and are therefore not acute angles.

Therefore a counterexample showing that an angle bisector will not always create two acute angles is when an angle bisector bisects angle 200 degrees. (see figure 2 in the attachment).

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