Chapter 5Relationships within Triangles

5.1 Midsegment Theorem and Coordinate Proof

5.2 Use Perpendicular Bisectors

5.3 Use Angle Bisectors of Triangles

5.4 Use Medians and Altitudes

5.5 Use Inequalities in a Triangle5.6 Inequalities in Two Triangles and Indirect Proof

Name _________________________________ Block _______

SOL G.5

The student, given information concerning the lengths of sides and/or measures of

angles in triangles, will:

a) order the sides by length, given the angle measures;

b) order the angles by degree measure, given the side lengths;

c) determine whether a triangle exists; and

d) determine the range in which the length of the third side must lie.

These concepts will be considered in the context of real-world situations.

TriangleMidsegment

TriangleMidsegment

A midsegment of a triangle is a segment

connecting the _____________________ of

two sides of the triangles.

Example: ______________

If a segment joins the midpoints of two sides of a triangle, then the

segment is _________________________ to the third side and

_______________ as long.

Using the diagram above, if BD is a midsegment of ∆ACE, then:

1) _______________________ 2) __________________________

Practice: D is the midpoint of AB and E is the midpoint of BC.

A

D

B

E

C

1) If DE = 8, find AC.

2) If AC = 17, find DE.

3) If BC = 9, find EC.

4) If DB = 5, find AB.

5) If m∠BDE = 35˚, find m∠BAC.

6) If m∠BCA = 110˚, find m∠DEB.

Practice: DE is a midsegment of ∆ABC. Find the value of x.

1. 2. 3.

5.1 Midsegment Theorem and Coordinate Proof

Use ∆GHJ, where D, E and F are midpoints of the sides.

1. If DE = 4x + 5 and GJ = 3x + 25, find DE.

2. If EF = 2x + 7 and GH = 5x – 1, what is EF?

You try!

Use ∆ABC, the midpoints are L, M, and N.

LM ∥ ____________ AB ∥ ____________

If AC = 14, then LN = ______________

If MN = 8, then AB = ______________

If NC = 3, then LM = _______________

If LN = 5, then ___________ = 10

If LM = 3x + 1 and BC = 10x – 6, then LM = ___________________

If NM = x – 1 and AB = 3x – 7, then AB = __________________

5.2 Use Perpendicular Bisectors

Perpendicular Bisector

Perpendicular Bisector

A line segment that is perpendicular (creates right angles) to a

side of a triangle and passes through its _______________________.

EquidistantTwo points are the ________________________________ from a third

point, creating congruent segments.

If a point is on the perpendicular bisector of a segment, then it is

___________________________ from the endpoints of the segment.

If ________ is the bisector of ________

then ___________________.

Perpendicular Bisector Theorem

If a point is equidistant from the endpoints of a

segment, then the point lies on the

____________________________________.

If __________________ then

_______________________________.

Example: KM is the perpendicular bisector of JL. Find JK and ML.

Example: DC is the perpendicular bisector of AB and AE ≌ BE.

What segment lengths are equal?

Is E on DC? Explain.

Concurrency

Concurrency of Perpendicular Bisectors of a

Triangle

The perpendicular bisectors of a triangle intersect at a point

that is ______________________________ from the vertices of

the triangle.

If PD, PE, and PF are perpendicular

bisectors, then ____________________.

A point of concurrency is the point where three or more lines,

rays, or segments intersect.

The point of intersection of the lines, rays, or segments is

called the ____________________________________________.

In the diagram, the perpendicular bisectors of ∆ABC meet at point G. Find the indicated

measure.

1. Find BG. 2. Find GA.

You try!

1. CD is the perpendicular bisector of AB.

A. What is the relationship between AD and DB?

B. What is the relationship between ∠ADC and ∠BDC?

C. What is the relationship between AC and CB? Explain.

2. In ∆DEG, FH is a perpendicular bisector of DG with H on DG. If DH = 2y + 3, GH = 7y – 42,

and m∠FHG = (x2 + 9)˚, find the value of x and y.

What is the measure of DG?

For questions 3 – 4, circle the correct word or phrase to complete each sentence.

3. Three or more lines that intersect at a common point are called _______________________

lines.

A. Parallel B. Perpendicular C. Concurrent

4. Any point on the perpendicular bisector of a segment is ______________________________

the endpoints of the segment.

A. Parallel to B. Equidistant to C. Congruent to

D

F

H G

E

5.3 Use Angle Bisectors of a Triangle

Recall, an angle bisector is a ray that divides an angle into two congruent adjacent angles.

Angle Bisector

Angle Bisector Theorem

If a point is on the bisector of an angle, then it

is ____________________from the two sides of

the angle.

Since AD is an bisector, then _________________.

If a point is in the interior of an angle and is equidistant from the

sides of the angle, then it lies on the bisector of the angle.

Can you conclude that BD bisects ∠ABC? Explain.

Find the value of x.

4. 5.

1. 2. 3.

Concurrency of Angle Bisectors of a Triangle:

The angle bisectors of a triangle intersect at a

point that is equidistant from the sides of the

triangle.

You try!

Find the value of x.

1. 2. 3.

4. Write an equation of the perpendicular bisector of the segment with endpoints P(-2, 3) and

Q(4, 1). Show all work.

Midpoint of PQ _____________

SlopePQ __________

Slope⊥ _______________

Y-Intercept ⊥ ________________

Equation of Perpendicular Bisector ____________________

30˚

30˚

10

x + 3

(6x + 14)˚

(9x – 1)˚

3x 4x – 9

5.4 Medians and Altitudes

MedianA median of a triangle is a segment from a _______________ to the

_____________________ of the opposite side.

Each triangle has three medians! The medians of a triangle are

concurrent, and the point of concurrency is called the _________________

inside the triangle. Medians do NOT have to be perpendicular!

Concurrency of Medians

The medians of a triangle intersect at a point that is two thirds of the

distance from each vertex to the midpoint of the opposite side.

If P is the centroid of ∆ABC, then AP = AD, BP = BF, CP = CE 23

23

23

Example: P is the centroid of ∆DEF, EH ⊥ DF, DH = 9, DG = 7.5, EP = 8, and DE = FE.

A. Find the length of FH.

B. Find the length of EH.

C. Find the length of PH.

D. Find the perimeter of ∆DEF.

E

J

F

G

DH

P

8

7.5

9

Example: Point G is the centroid of ∆ABC. Use the given information to find the value of x.

FG = x + 8 and GA = 6x – 4

Altitude

**Altitude is often called height

An altitude is a perpendicular segment that connects the

____________________________ to its __________________________.

Altitudes are NOT bisectors! And, they do not have to be inside the ∆.

Concurrency of Altitudes

The lines containing the altitudes of

a triangle are concurrent.

In ∆ABC, is BD a perpendicular bisector, median or altitude?

1. 2. 3.

4. Use the picture to name the following parts of the triangle.

A. Median

B. Angle Bisector

C. Perpendicular Bisector

D. Altitude

You try!

1. Use the picture to name the following parts of the triangle.

A. Median

B. Angle Bisector

C. Perpendicular Bisector

D. Altitude

2. ZC is an altitude, CYW = 9x + 38, and WZC = 4x.

Solve for x.

Find the measures of ∠CZW and ∠CYW.

.5. XW is an angle bisector. ∠YXZ = 20y + 8, ∠WXY = 8y + 12, ∠XZY = 12y. Label the diagram.

A. Solve for y.

B. Find the measure of ∠ZWX.

C. Is XW an altitude? Justify your answer.

Is it a Triangle?

Triangle Inequalities:

• Allow us to order sides or angles of triangles by length

• Do not give us exact lengths; just relative lengths (order)

• Help us to determine whether triangles can be formed

Determine if the following side

lengths could form a triangle. Prove your

answer with an inequality.

Hint: the sum of the two smaller numbers

must be larger than the third number.

1. 8, 17, 24

3. 3, 3, 7

5. 28, 50, 22

2. 24, 12, 11

4. 35, 41, 7

5. 6, 18, 14

Finding a third side

rangeLet x = third side

Given two sides of a triangle, you can set

up an inequality using the sum and difference to show range of possible

lengths for the third

side.

6. 14 and 22

8. 3 and 11

7. 31 and 28

9. 19 and 45

5.5 Use Inequalities in a Triangle

Example: If a triangle has lengths of 15 feet and 27 feet, check all possible lengths for

the third side.

34 ft 12 ft 29 ft 18ft 43 ft 15 ft 52 ft

Ordering Angles

The angles of a triangle can be put in order by comparing the sides.

The smallest angle is always opposite the _____________________________.

The largest angle is always opposite the _____________________________.

Ordering Sides

The sides of a triangle can be put in order by comparing the angles.

The shortest side is always opposite the _____________________________.

The longest side is always opposite the _____________________________.

Largest angle

Largest side

Smallest angle

Smallest side

List the sides and the angles in order from smallest to largest.

1. 2.10

79

RS

T

33˚A B

C

You try!

1. Sketch and label the triangle described.

Side lengths: about 3 m, 7 m, and 9 m, with longest side on the bottom

Angle measures: 16˚, 41˚, and 123˚, with the smallest angle on the left.

2. Is it possible to construct a triangle with the given side lengths? If not, explain why not.

A. 6, 7, 11 B. 3, 6, 9 C. 28, 34, 39

3. Describe, using an inequality, the 4. Describe the lengths of the third side

possible values of x. of the triangle given the lengths of the

other two sides.

A. A. 3, 4 B. 12, 18

5. Find the value of x and use the indicated angle measures to list the sides of ABC in order

from shortest to longest.

m∠A = (9x + 29)˚, m ∠ B = (93 – 5x)˚, and m ∠ C = (10x + 2)˚

x + 112x + 10

5x – 9

5.6 Inequalities in Two Triangles and Indirect Proof

The hinge theorem states that if two sides of one triangle are

congruent to two sides of another triangle, and the included angle

of the first is larger than the included angle of the second, then the

third side of the first triangle is _________________ than the third side

of the second triangle

Hinge Theorem

Hinge Theorem

If two sides of one triangle are congruent to two sides of another

triangle, and the third side of the first is longer than the third side of

the second, then the included angle of the first is __________________

than the included angle of the second.

WX ______ ST

∠C ______ ∠F

Complete each statement with <, > or =.

1. 2. 3.

Write an inequality to describe the restrictions on the value of x.

4. 5.

A

C B

D F

E

105˚

110˚

AB ______ DE

6 6

21

m∠1 ______ m∠2

1

2

8 7

m∠1 ______ m∠2

6

6

x

7

65˚

70˚

12

12

3x + 2

12x – 7

Steps for writing an indirect proof

1.

2.

3.

Identify the statement you want to prove. Assume temporarily that this statement is ____________ by assuming that its opposite is true.

Reason logically until you reach a __________________________.

Point out that the desired conclusion must be ___________ because the contradiction proves the temporary assumption

false.

In other words, you are proving something indirectly by showing that is cannotbe false.

Example 1: Suppose you want to write indirect proof of this statement: “If x is an odd number, then x is not divisible by 4.” What temporary assumption should start your proof?

Example 2: Suppose you want to write indirect proof of this statement: “If x + y ≠ 14, and y = 5, then x ≠ 9.”What temporary assumption should start your proof?

You try!

Complete each statement with <, > or =.

1. 2. 3.

Write an inequality to describe the restrictions on the value of x.

4. 5.

6. Match the conclusion on the right with the given information on the left.

1. AB = BC, m∠1 > m∠2 a. m∠7 > m∠8

2. AE > EC, AD = CD b. AD > AB

3. m∠9 < m∠10, BE = ED c. m∠3 + m∠4 = m∠5 + m∠6

4. AB = BC, AD = CD d. AE > EC

FG ______ LM m∠1 ______ m∠2 m∠1 ______ m∠2

3

3

x + 3

3x + 2

110˚ 27˚

4x – 3

2xA

D

B C