ANGLE BISECTOR, MIDSEGMENT, & SIDE-SPLITTER THEOREMS5.4

WARM UP

• Solve for x:

•3

5=

𝑥

25

•6

𝑥=

36

72

•𝑥

9=

4

𝑥

LESSON 5.4: ANGLE SPLITTER, SIDE SPLITTER, MIDSEGMENT 2

OBJECTIVES

To understand and use the Angle Bisector, Midsegment and Side-Splitter Theorems

LESSON 5.4: ANGLE SPLITTER, SIDE SPLITTER, MIDSEGMENT 3

ANGLE BISECTOR

ANGLE BISECTOR is created when a line (the bisector) passes through a vertex of an angle, creating two equal smaller angles.

LESSON 5.4: ANGLE SPLITTER, SIDE SPLITTER, MIDSEGMENT 4

ANGLE BISECTOR

1) Find 𝑥 if 𝑚∠1 = 6 + 3𝑥 and 𝑚∠2 = 4𝑥 − 4. 𝐸𝑃 is an angle bisector of ∠𝐸.

LESSON 5.4: ANGLE SPLITTER, SIDE SPLITTER, MIDSEGMENT 5

ANGLE BISECTOR

2) Find 𝑥 if 𝑚∠1 = 2𝑥 + 10 and 𝑚∠2 = 3𝑥 + 3. 𝑃𝑊 is an angle bisector of ∠𝑊.

LESSON 5.4: ANGLE SPLITTER, SIDE SPLITTER, MIDSEGMENT 6

ANGLE BISECTOR

3) Find 𝑥 if 𝑚∠1 = 25𝑥 + 1 and 𝑚∠𝐴𝐶𝐵 = 51𝑥 + 1. 𝑃𝐶 is an angle bisector of ∠𝐶.

LESSON 5.4: ANGLE SPLITTER, SIDE SPLITTER, MIDSEGMENT 7

ANGLE BISECTOR THEOREM

ANGLE BISECTOR THEOREMIf an angle of a triangle is bisected, then the angle bisector divides the opposite side of the triangle into two segments that are proportional to the other two sides of the triangle.

LESSON 5.4: ANGLE SPLITTER, SIDE SPLITTER, MIDSEGMENT 8

ÐABD @ ÐDBC

AD

DC=

BA

BCtherefore

ANGLE BISECTOR THEOREM

4) Find the missing length indicated.

LESSON 5.4: ANGLE SPLITTER, SIDE SPLITTER, MIDSEGMENT 9

ANGLE BISECTOR THEOREM

5) Find the missing length indicated.

LESSON 5.4: ANGLE SPLITTER, SIDE SPLITTER, MIDSEGMENT 10

ANGLE BISECTOR THEOREM

6) Solve for 𝑥.

LESSON 5.4: ANGLE SPLITTER, SIDE SPLITTER, MIDSEGMENT 11

ANGLE BISECTOR THEOREM

•7) Solve for 𝑥.

LESSON 5.4: ANGLE SPLITTER, SIDE SPLITTER, MIDSEGMENT 12

SIDE-SPLITTER THEOREM

LESSON 5.4: ANGLE SPLITTER, SIDE SPLITTER, MIDSEGMENT 13

If a line is parallel to one side of a triangle and intersects other two sides of triangle, then it will divide other two sides proportionally.

Such that: 𝑎

𝑏=

𝑐

𝑑

Or 𝑎

𝑎+𝑏=

𝑐

𝑐+𝑑

SIDE-SPLITTER THEOREM

LESSON 5.4: ANGLE SPLITTER, SIDE SPLITTER, MIDSEGMENT 14

• Solve for x:

SIDE-SPLITTER THEOREM

• Solve for x:

LESSON 5.4: ANGLE SPLITTER, SIDE SPLITTER, MIDSEGMENT 15

MIDSEGMENT

•MIDSEGMENTS are line segments created by connecting the midpoints of two sides of a triangle.

LESSON 5.4: ANGLE SPLITTER, SIDE SPLITTER, MIDSEGMENT 16

MIDSEGMENT THEOREM

•A MIDSEGMENT of a triangle is parallel to the third side and is half as long.

LESSON 5.4: ANGLE SPLITTER, SIDE SPLITTER, MIDSEGMENT 17

MIDSEGMENT THEOREM

9) Find 𝐽𝐾. 10) Find 𝐷𝐵.

LESSON 5.4: ANGLE SPLITTER, SIDE SPLITTER, MIDSEGMENT 18

MIDSEGMENT THEOREM

11) Find 𝑇𝑉. 12) Find 𝑋𝑊.

LESSON 5.4: ANGLE SPLITTER, SIDE SPLITTER, MIDSEGMENT 19

MIDSEGMENT THEOREM

13) Find the lengths of 𝐵𝐶 and 𝑌𝑍.

LESSON 5.4: ANGLE SPLITTER, SIDE SPLITTER, MIDSEGMENT 20

HOMEWORK

•Worksheet 5.4+Review

LESSON 5.4: ANGLE SPLITTER, SIDE SPLITTER, MIDSEGMENT 21