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