Angle Bisector Proportionality

Question 1: State the Angle Bisector Theorem.

• The angle bisector of a triangle has no effect on the opposite side.

• The angle bisector of a triangle divides the opposite side into segments that are proportional to the other two sides of the triangle.

• The angle bisector of a triangle divides the opposite side into segments that are equal to the other two sides of the triangle.

• The angle bisector of a triangle divides the opposite side into segments that are random lengths.

Question 3: In triangle XYZ, if XZ is the angle bisector of angle YXW, and XY = 8, YZ = 10, find the length of XZ.

• 12

• 5

• 6

• 4

Question 4: Prove that the angle bisector divides the opposite side into segments that are proportional to the adjacent sides.

• Pythagorean theorem

• Law of Sines

• Law of Cosines

• Angle bisector theorem

Question 5: What is the definition of proportional segments in a triangle?

• Segments divided by a line parallel to one side of the triangle, creating proportional lengths on the other two sides

• Segments divided by a line intersecting the triangle at a random angle

• Segments divided by a line parallel to the hypotenuse of the triangle

• Segments divided by a line perpendicular to one side of the triangle

Question 6: In triangle PQR, if PS is the angle bisector of angle P, and PQ = 12, PR = 15, find the length of PS.

• 6

• 10

• 9

• 13

Question 7: If a line divides two sides of a triangle proportionally, what can you say about that line?

• Angle bisector

• Median

• Perpendicular bisector

• Altitude

Question 8: How can you apply angle bisector proportionality to find unknown side lengths in a triangle?

• Use the Pythagorean theorem to find the unknown side lengths.

• Count the number of angles in the triangle to determine the unknown side lengths.

• Apply the Law of Sines to calculate the unknown side lengths.

• Use the angle bisector theorem to set up and solve proportions to find the unknown side lengths.

Question 9: In triangle ABC, if AD is the angle bisector of angle A, and AB = 10, AC = 14, find the length of AD.

• 12

• 5

• 3

• 7

Question 10: If a line divides two sides of a triangle proportionally, what can you conclude about the angles formed by that line with the two sides?

• Acute

• Congruent

• Parallel

• Perpendicular