Question

question 17 of 25

which of the following is the angle bisector theorem?

a. if a point is in the interior of an angle and proportional from the two sides of the angle, then it lies on the bisector of the angle.

b. if a point is on the bisector of an angle, then it is proportional from the two sides of the angle.

c. if a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.

d. if a point is in the interior of an angle and equidistant from the two sides of the angle, then it lies on the bisector of the angle.

Explanation:

Define the Angle Bisector Theorem

Using the Angle Bisector Theorem knowledge point:

• Hypothesis: A point lies on the bisector of an angle.
• Conclusion: The point is equidistant from the two sides of the angle.

Analyze the given options

• Option A describes a proportional relationship for a point in the interior, which is mathematically incorrect.
• Option B mentions a proportional relationship, which is incorrect because the actual relationship is equality of distance (equidistant).
• Option C states: "If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle." This matches the definition of the theorem.
• Option D describes the converse of the theorem: "If a point is in the interior of an angle and equidistant from the two sides of the angle, then it lies on the bisector of the angle."

Answer:

• (A) If a point is in the interior of an angle and proportional from the two sides of the angle, then it lies on the bisector of the angle.
• (B) If a point is on the bisector of an angle, then it is proportional from the two sides of the angle.
• (C) If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. (Correct answer)
• (D) If a point is in the interior of an angle and equidistant from the two sides of the angle, then it lies on the bisector of the angle.