FeatureLesson Geometry Lesson Main In GHI, R, S, and T are midpoints. 1. Name all the pairs of parallel segments. 2. If GH = 20 and HI = 18, find RT. 3.

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GeometryGeometry

LessonMain

In GHI, R, S, and T are midpoints.

1. Name all the pairs of parallel segments.

2. If GH = 20 and HI = 18, find RT.

3. If RH = 7 and RS = 5, find ST.

4. If m G = 60 and m I = 70, find m GTR.

5. If m H = 50 and m I = 66, find m ITS.

6. If m G = m H = m I and RT = 15, find the perimeter of GHI.

RT || HI, RS || GI, ST || HG

90

64

70

7

9

Lesson 5-1 Quiz – Midsegments of Triangles

5-1

Lesson 5-2

Bisectors in TrianglesBisectors in Triangles

FeatureLesson

GeometryGeometry

LessonMain

Lesson 5-2

Bisectors in TrianglesBisectors in Triangles

5-2

When a point is the same distance from two or moreobjects, the point is said to be equidistant fromthe objects. Triangle congruence theorems can beused to prove theorems about equidistant points.

FeatureLesson

GeometryGeometry

LessonMain

Lesson 5-2

Bisectors in TrianglesBisectors in Triangles

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FeatureLesson

GeometryGeometry

LessonMain

Lesson 5-2

Bisectors in TrianglesBisectors in Triangles

5-2

FeatureLesson

GeometryGeometry

LessonMain

Lesson 5-2

Bisectors in TrianglesBisectors in Triangles

5-2

The shortest segment from a point to a line is perpendicular to the line. This fact is used to define the distance from a point to a line as the length of the perpendicular segment from the point to the line.

FeatureLesson

GeometryGeometry

LessonMain

Lesson 5-2

Bisectors in TrianglesBisectors in Triangles

5-2

FeatureLesson

GeometryGeometry

LessonMain

Use the map of Washington, D.C. Describe the set of points

that are equidistant from the Lincoln Memorial and the Capitol.

The Converse of the Perpendicular Bisector Theorem states If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

Lesson 5-2

Bisectors in TrianglesBisectors in Triangles

Quick Check

Additional Examples

5-2

Real-World Connection

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GeometryGeometry

LessonMain

(continued)

A point that is equidistant from the Lincoln Memorial and the Capitol

must be on the perpendicular bisector of the segment whose

endpoints are the Lincoln Memorial and the Capitol.

Therefore, all points on the perpendicular bisector of the segment

whose endpoints are the Lincoln Memorial and the Capitol are

equidistant from the Lincoln Memorial and the Capitol.

Lesson 5-2

Bisectors in TrianglesBisectors in Triangles

Additional Examples

5-2

Quick Check

FeatureLesson

GeometryGeometry

LessonMain

Find x, FB, and FD in the diagram above.

FD = FB Angle Bisector Theorem

7x – 37 = 2x + 5 Substitute.

7x = 2x + 42 Add 37 to each side.

5x = 42 Subtract 2x from each side.

x = 8.4 Divide each side by 5.

FB = 2(8.4) + 5 = 21.8 Substitute.

FD = 7(8.4) – 37 = 21.8 Substitute.

Lesson 5-2

Bisectors in TrianglesBisectors in Triangles

Additional Examples

5-2

Using the Angle Bisector Theorem

Quick Check

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GeometryGeometry

LessonMain

Use this figure for Exercises 1–3.

1. Find BD.

2. Complete the statement: C is equidistant from ? .

3. Can you conclude that CN = DN? Explain.

Use this figure for Exercises 4–6.

4. Find the value of x.

5. Find CG.

6. Find the perimeter of quadrilateral ABCG.

6

Lesson 5-2

points A and B

16

48

8

No; if CN = DN, CNB DNB by SAS and CB = DB by CPCTC, whichis false.

Bisectors in TrianglesBisectors in Triangles

Lesson Quiz

5-2

FeatureLesson

GeometryGeometry

LessonMain

(For help, go to Lesson 1-7.)

Lesson 5-2

Bisectors in TrianglesBisectors in Triangles

1. Draw a triangle, XYZ. Construct STV so that STV XYZ.

2. Draw acute P. Construct Q so that

3. Draw AB. Construct a line CD so that CD AB and CD bisects AB. 4. Draw acute angle E. Construct the bisector of E.

TM bisects STU so that m STM = 5x + 4 and m MTU = 6x – 2.

5. Find the value of x.

6. Find m STU.

Use a compass and a straightedge for the following.

Check Skills You’ll Need

Q P.

Check Skills You’ll Need

5-2

FeatureLesson

GeometryGeometry

LessonMain

Lesson 5-2

Bisectors in TrianglesBisectors in Triangles

Solutions

1. 2.

3. 4.

5. Since TM bisects STU, m STM = m MTU. So, 5x + 4 = 6x – 2. Subtract 5x from both sides: 4 = x – 2; add 2 to both sides: x = 6.

6. From Exercise 5, x = 6. m STU = m STM + m MTU = 5x + 4 + 6x – 2 = 11x + 2 = 11(6) + 2 = 68.

1-4. Answers may vary. Samples given:

Check Skills You’ll Need

5-2