Exploring Triangle Angle Bisector Theorem and Side Splitter Corollary

This lesson covers the triangle bisector theorem, explaining how a bisected angle divides the opposite side into proportional segments. Several examples demonstrate solving for missing side lengths and total side lengths using proportions. The lesson also introduces the side splitter theorem, which states that a line parallel to one side of a triangle divides the other two sides proportionally. Examples illustrate solving for unknowns using this theorem.

What does the Triangle Bisector Theorem state?

• A ray bisecting a triangle divides the opposite side into segments of the same length.

• A ray bisecting a triangle divides the opposite side into proportional segments.

• A ray bisecting a triangle divides the opposite side into unequal segments.

• A ray bisecting a triangle divides the opposite side into equal segments.

In the first example, if the bisected angle makes 6 proportional to the missing side length, what is the missing side length?

• 18

• 9

• 3

• 2

In the second example, what is the total length of the side if the missing piece is found to be 8?

• 10

• 16

• 20

• 18

In the third example, if the full side length is 12 and one piece is 4, what is the length of the other piece?

• 10

• 6

• 4

• 8

What is the value of x if 4 to 2 is proportional to 2 to 2x - 9?

• 7

• 6

• 5

• 4

What does the Side Splitter Theorem state?

• A parallel line to one side of a triangle divides the other two sides into unequal segments.

• A parallel line to one side of a triangle divides the other two sides into segments of the same length.

• A parallel line to one side of a triangle divides the other two sides into proportional segments.

• A parallel line to one side of a triangle divides the other two sides into equal segments.

In the first example of the Side Splitter Theorem, if 4 over 3 equals x over 6, what is the value of x?

• 8

• 6

• 12

• 9

In the second example, if the large piece is 18 and the whole side is 12, what is the missing piece?

• 4

• 8

• 10

• 6

In the last example, if 9 over 6 equals x + 8 over 10, what is the value of x?

• 5

• 6

• 7

• 8

What is the final step in solving for x in the last example?

• Subtract 48 from both sides

• Add 48 to both sides

• Divide 42 by 6

• Multiply 42 by 6