5-3 Perpendicular and Angle BisectorsPerpendicular and Angle Bisectors

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Exit Ticket

Lesson Presentation

5-3 Perpendicular and Angle Bisectors

Warm Up #7

1. Is 𝐷𝐸 ≅ 𝐷𝐹 ? Explain.

3. Define an angle bisector in your own words.

E F

D

61o

58o

2. What is the value of x ?

4. Draw an angle bisector.

Yes; mF = 61o byConverse of the Isosceles ∆ Thrm.

x = 4

An angle bisector divides an angle into two equal parts.

5-3 Perpendicular and Angle Bisectors

You hang a bulletin board over your desk using string. The bulletin board is crooked. When you straighten the bulletin board:

• What type of triangle does the string form with the top of the board? How do you know?

Visualize the vertical line along the wall that passes through the nail.

• What relationships exist between this line and the top edge of the straightened bulletin board? Explain.

5-3 Perpendicular and Angle Bisectors

5-3 Perpendicular and Angle Bisectors

SWBAT

Construct a perpendicular bisector and make conjectures about the geometric relationship formed by the endpoints and points on the bisector.

Construct angle bisectors and make conjectures about geometric relationships formed by the bisector and sides of the angle.

By the end of today’s lesson,

Connect to Mathematical Ideas (1)(F)

5-3 Perpendicular and Angle Bisectors

equidistant

locus

Vocabulary

5-3 Perpendicular and Angle Bisectors

Explore:

5-3 Perpendicular and Angle Bisectors

Explore: Make the conjecture

5-3 Perpendicular and Angle Bisectors

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.

5-3 Perpendicular and Angle Bisectors

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.

5-3 Perpendicular and Angle Bisectors

5-3 Perpendicular and Angle Bisectors

Given

7) 7) CPCTC

8)

Example 1:

8) Definition of Congruence

5-3 Perpendicular and Angle Bisectors

5-3 Perpendicular and Angle Bisectors

Example 2: Applying the Perpendicular Bisector Theorem and Its Converse

𝑩𝑫 is the perpendicular bisector of 𝑨𝑪, so B is equidistant from A and C .

𝐴𝐵 = 𝐵𝐶 Bisector Theorem.

4x = 6x – 10

2x = 10

x = 5

Combine liked terms

Divide both sides by 2.

Substitute the given values.

𝐴𝐵 = 4x

Now find:

4 (5) = 20

5-3 Perpendicular and Angle Bisectors

Example 3: Applying the Perpendicular Bisector Theorem and Its Converse

A park director wants to build a T-shirt stand equidistant from the Rollin’s Coaster and the Spaceship Shoot. What are the possible locations of the stand? Explain.

To be equidistant from the two rides, the stand should be on the perpendicular bisector of the segment connecting the rides. Find the midpoint A of 𝑅𝑆 and draw l through A perpendicular to 𝑅𝑆. The possible locations of the stand are all the points on line l.

5-3 Perpendicular and Angle Bisectors

Got It ? Solve With Your Partner

Problem 1 Applying Perpendicular Bisector

A park director wants to build a T-shirt stand equidistant from the Paddle boats and the Spaceship Shoot. What are the possible locations? Explain.

Any locus point on the perpendicular bisector of 𝑃𝑆.

5-3 Perpendicular and Angle Bisectors

5-3 Perpendicular and Angle Bisectors

5-3 Perpendicular and Angle Bisectors

Example 4: Applying the Angle Bisector Theorem and Its Converse

𝑅𝑀 = 𝑅𝑃 Bisector Theorem.

7x = 2x + 25

5x = 25

x = 5

Combine liked terms

Divide both sides by 5.

Substitute the given values.

𝑅𝑃 = 7x Now find: 7 (5) = 35

5-3 Perpendicular and Angle Bisectors

Got It ? Solve With Your Partner

Problem 2 Applying Angle Bisectors

𝐹𝐵= 𝐹𝐷 Bisector Theorem.

6x + 3 = 4x + 9

2x = 6

x = 3

Combine liked terms

Divide both sides by 3.

Substitute the given values.

𝐹𝐵 = 6x + 3

Now find:

6(3) + 3 = 21

What is the length of 𝑭𝑩 ?

5-3 Perpendicular and Angle Bisectors

Closure: Communicate Mathematical Ideas (1)(G)

What statement describes the points on the perpendicular

bisector of a segment ?

What statement can be made about a point on the bisector of an angle?

What are the similarities and differences between an angle bisector and a perpendicular bisector?

It is equidistant from the sides of the angle.

They are equidistant from the endpoints of the segment.

5-3 Perpendicular and Angle Bisectors

Exit Ticket: Apply Mathematics (1)(A)

Use the diagram for Items 1–2.

1. Given that mABD = 16°, find mABC.

2. Given that mABD = (2x + 12)° andmCBD =(6x – 18)°, find mABC.

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