Question

When −−→BX bisects 6 ABC, 6 ABX ∼= 6 CBX.

One student claims there is always a related equation m6 ABX =1/2 ∠ABC.
Another student claims the related equation 2m∠ ABX = m ∠ ABC.
Who is correct? Explain.

Answer 1

The correct relation when ––→BX bisects ∠ABC is m∠ABX = 1/2 ∠ABC, meaning the bisector creates two congruent angles each half of ∠ABC. 2m∠ABX = m∠ABC is incorrect as it contradicts the principle of a bisector.

Step-by-step explanation:

When the segment ––→BX bisects ∠ABC, it means that ∠ABX and ∠CBX are congruent. Since △ABX ≅ △CBX, they are similar triangles which implies that their corresponding angles are equal. Therefore, m∠ABX = m∠CBX. The claim that m∠ABX = 1/2 ∠ABC is the correct one as the bisector creates two angles (∠ABX and ∠CBX) that are each half of the total angle ∠ABC.

Contradicting the incorrect claim, it's not true that 2m∠ABX = m∠ABC since this equation would imply that ∠ABX is half of ∠ABC, which is impossible if ––→BX is a bisector. Thus, m∠ABX = 1/2 ∠ABC accurately describes the relationship between the angles when a segment bisects an angle of a triangle.