1. 5-3 5-1 Perpendicular and Angle Bisectors Warm Up Lesson Presentation
Lesson Quiz
Holt Geometry

2. 5-3
Warm up:

3. 5-3
Warm up:

4. 5-3

5. 5-3
Midpoint Formula:
Slope formula:

6. 5-3 Point-Slope form: Use when you are given TWO points
Slope-intercept form:
Use when you are given TWO points
Fill in where the red circles are

7. 5-3
Find the midpoint and the slope of the segment (3,-2) and (-4,1)

8. 5-3
Write the equation of a line that passes through (8,-4) and (5,2).

9. 5-3
Objective:
Prove and apply theorems about perpendicular bisectors; Prove and apply theorems about angle bisectors

10. 5-3 When a point is the same distance from two or more
objects, the point is said to be equidistant from
the objects. Triangle congruence theorems can be
used to prove theorems about equidistant points.

11. 5-3 Perpendicular bisector theorem: X A Y B
The triangles are congruent by SAS
CPCTC allows me to conclude that

12. 5-3

13. 5-3
A locus is a set of points that satisfies a given condition. The perpendicular bisector of a segment can be defined as the locus of points in a plane that are equidistant from the endpoints of the segment.

14. 5-3

15. 5-3
Example 1: Applying the Perpendicular Bisector Theorem and Its Converse
Find each measure. MN
MN = LN
 Bisector Thm.
MN = 2.6
Substitute 2.6 for LN.

16. 5-3
Example 2: Applying the Perpendicular Bisector Theorem and Its Converse
Find each measure. BC
Since AB = AC and , is the perpendicular bisector of by the Converse of the Perpendicular Bisector Theorem.
BC = 2CD
Def. of seg. bisector.
BC = 2(12) = 24
Substitute 12 for CD.

17. 5-3
Example 3: Applying the Perpendicular Bisector Theorem and Its Converse
Find each measure. TU
TU = UV
 Bisector Thm.
3x + 9 = 7x – 17
Substitute the given values.
9 = 4x – 17
Subtract 3x from both sides.
26 = 4x
Add 17 to both sides.
6.5 = x
Divide both sides by 4.
So TU = 3(6.5) + 9 = 28.5.

18. 5-3

19. 5-3
Based on these theorems, an angle bisector can be defined as the locus of all points in the interior of the angle that are equidistant from the sides of the angle.

20. 5-3 Angle bisector theorem: X v B A v Y
The triangles are congruent by AAS
CPCTC allows me to conclude that

21. 5-3 Angle bisector theorem: X v B A v Y
The triangles are congruent by HL
CPCTC allows me to conclude that

22. Example 1: Applying the Angle Bisector Theorem
5-3
Example 1: Applying the Angle Bisector Theorem
Find the measure. BC
BC = DC
 Bisector Thm.
BC = 7.2
Substitute 7.2 for DC.

23. Example 2: Applying the Angle Bisector Theorem
5-3
Example 2: Applying the Angle Bisector Theorem
Find the measure.
mEFH, given that mEFG = 50°.
Since EH = GH,
and , bisects
EFG by the Converse
of the Angle Bisector Theorem.
Def. of  bisector
Substitute 50° for mEFG.

24. Example 2C: Applying the Angle Bisector Theorem
5-3
Example 2C: Applying the Angle Bisector Theorem
Find mMKL.
, bisects JKL
Since, JM = LM, and
by the Converse of the Angle
Bisector Theorem.
mMKL = mJKM
Def. of  bisector
3a + 20 = 2a + 26
Substitute the given values.
a + 20 = 26
Subtract 2a from both sides.
a = 6
Subtract 20 from both sides.
So mMKL = [2(6) + 26]° = 38°

25. 5-3 Check It Out! Example 2a Given that YW bisects XYZ and
WZ = 3.05, find WX.
WX = WZ
 Bisector Thm.
So WX = 3.05

26. 5-3
Check It Out! Example 2b
Given that mWYZ = 63°, XW = 5.7, and ZW = 5.7, find mXYZ.
mWYZ + mWYX = mXYZ
 Bisector Thm.
mWYZ = mWYX
Substitute m WYZ for mWYX .
mWYZ + mWYZ = mXYZ
2mWYZ = mXYZ
Simplify.
2(63°) = mXYZ
Substitute 63° for mWYZ .
126° = mXYZ
Simplfiy .

27. Example 4: Writing Equations of Bisectors in the Coordinate Plane
5-3
Example 4: Writing Equations of Bisectors in the Coordinate Plane
Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints C(6, –5) and D(10, 1).
Step 1 Graph .
The perpendicular bisector of is perpendicular to at its midpoint.

28. 5-3 Example 4 Continued Step 2 Find the midpoint of .
Midpoint formula.
mdpt. of =

29. 5-3
Example 4 Continued
Step 3 Find the slope of the perpendicular bisector.
Slope formula.
Since the slopes of perpendicular lines are opposite reciprocals, the slope of the perpendicular bisector is

30. 5-3
Example 4 Continued
Step 4 Use point-slope form to write an equation. The perpendicular bisector of has slope and passes through (8, –2).
y – y1 = m(x – x1)
Point-slope form
Substitute –2 for y1, for m, and 8 for x1.

31. 5-3
Example 4 Continued

32. 5-3
Check It Out! Example 4
Write an equation in slope-intercept form for the perpendicular bisector of the segment with endpoints P(5, 2) and Q(1, –4).
Step 1 Graph PQ.
The perpendicular bisector of is perpendicular to at its midpoint.

33. Check It Out! Example 4 Continued
5-3
Check It Out! Example 4 Continued
Step 2 Find the midpoint of PQ.
Midpoint formula.

34. Check It Out! Example 4 Continued
5-3
Check It Out! Example 4 Continued
Step 3 Find the slope of the perpendicular bisector.
Slope formula.
Since the slopes of perpendicular lines are
opposite reciprocals, the slope of the
perpendicular bisector is

35. Check It Out! Example 4 Continued
5-3
Check It Out! Example 4 Continued
Step 4 Use point-slope form to write an equation.
The perpendicular bisector of PQ has slope and passes through (3, –1).
y – y1 = m(x – x1)
Point-slope form
Substitute.

36. 5-3 Lesson Quiz: Part I Use the diagram for Items 1–2.
1. Given that mABD = 16°, find mABC.
2. Given that mABD = (2x + 12)° and mCBD = (6x – 18)°, find mABC.
32°
54°
Use the diagram for Items 3–4.
3. Given that FH is the perpendicular bisector of EG, EF = 4y – 3, and FG = 6y – 37, find FG.
4. Given that EF = 10.6, EH = 4.3, and FG = 10.6, find EG. 65
8.6

37. 5-3
Lesson Quiz: Part II
5. Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints X(7, 9) and Y(–3, 5) .