Chapter 5: Relationships in Triangles

Lesson 5.1

Bisectors, Medians, and Altitudes

Perpendicular Bisector

Definition Facts to Know Point of Concurrency

Example

A line, segment or ray that passes through the midpoint of the opposite side and is perpendicular to that side

Any point on a perpendicular bisector is equidistant from the endpoints

Circumcenter:

The point where 3 perpendicular bisectors intersect

- the circumcenter is equidistant from all vertices of the triangle

D CB

A

BD = CD

AD BC

E is the circumcenter-

AE = BE = CE

E

Median

Definition Facts to Know Point of Concurrency

Example

A segment that goes from a vertex of the triangle to the midpoint of the opposite side

The median splits the opposite side into two congruent segments

Small = 1/3 median

Big = 2/3 median

2 x small = big

Centroid:

The point where 3 medians intersect

D CB

A

BD = CD

E is the centroid-

ED = 1/3 AD

AE = 2/3 AD

2 ED = AE

E

Angle Bisector

Definition Facts to Know Point of Concurrency

Example

A line, segment, or ray that passes through the middle of an angle and extends to the opposite side

Any point on an angle bisector is equidistant from the sides of the triangle

Incenter:

The point where 3 angle bisectors intersect

-the incenter is equidistant from all sides of the triangle

D CB

A

FE

BAD = CAD

G is the incenter-

EG = FG

G

Altitude

Definition Facts to Know Point of Concurrency

Example

A segment that goes from a vertex of the triangle to the opposite side and is perpendicular to that side

Orthocenter:

The point where 3 altitudes intersect

D CB

A

AD BC

C. Find the measure of EH.

A. Find QS.

B. Find WYZ.

In the figure, A is the circumcenter of ΔLMN. Find x if mAPM = 7x + 13.

In the figure, point D is the incenter of ΔABC. What segment is congruent to DG?

In ΔXYZ, P is the centroid and YV = 12. Find YP and PV.

In ΔLNP, R is the centroid and LO = 30. Find LR and RO.

Lesson 5.2

Inequalities and Triangles

Foldable

Fold the paper into three sections (burrito fold) Then fold the top edge down about ½ an inch

Unfold the paper and in the top small rectangles label each column…

Exterior Angle Inequality Inequality with Sides Inequality with Angles

•Exterior Angle =

Remote Int. + Remote Int.

-The exterior angle is greater than either of the remote interior angles by themselves

rem. Int. < ext.

Ex:

-The biggest side is across from the biggest angle-The smallest side is across from the smallest angle

Ex:

-The biggest angle is across from the biggest side/ the smallest angle is across from the smallest side

Ex:

List the angles of ΔABC in order from smallest to largest.

List the sides of ΔABC in order from shortest to longest.

What is the relationship between the lengths of RS and ST?

______

What is the relationship between the

measures of A and B?

Lesson 5.4

The Triangle Inequality

Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle is greater than the length of the third side

Triangle Inequality Theorem Problems Determine if the measures

given could be the sides of a triangle. 16, 17, 19

16 + 17 = 33 yes, the sum of the two smallest sides is larger than the third side

6, 9, 15

6 + 9 = 15 no, the sum of the two smallest sides is equal to the other side so it cannot be a triangle

Find the range for the measure of the third side given the measures of two sides. 7.5 and 12.1

12.1- 7.5 < x < 12.1 + 7.5

4.6 < x < 19.6 9 and 41

41-9 < x < 41 + 9

32 < x < 50

Determine whether it is possible to form a triangle with side lengths 5, 7, and 8.

Is it possible to form a triangle with the given side lengths of 6.8, 7.2, 5.1? If not, explain why not.

Find the range for the measure of the third side of a triangle if two sides measure 4 and 13.

In ΔPQR, PQ = 7.2 and QR = 5.2. Which measure cannot be PR?

A 7

B 9

C 11

D 13

Lesson 5.3

Indirect Proof

Steps to Completing an Indirect Proof: Assume that ______________ (the

conclusion is false) Then _______________ (show that the

assumption leads to a contradiction) This contradicts the given information that ________________.

Therefore, __________________ (rewrite the conclusion) must be true.

B. State the assumption you would make to start an indirect proof for the statement 3x = 4y + 1.

Example Indirect Proof

Given: 5x < 25

Prove: x < 5x 5.1. Assume that

2. Then x= 9 And 5(9)= 45 45> 25

This contradicts the given info that 5x < 25

3. Therefore, x < 5 must be true.

Example Indirect ProofGiven: m is not parallel to n

Prove: m 3 m 2

2

3m

n

1. Assume that m 3 = m 2

2. Then, angles 2 and 3 are alternate interior anglesWhen alternate interior angles are congruent then the

lines that make them are parallel. This contradicts the given info that m is not parallel to n

3. Therefore, m 3 m 2 must be true.

Write an indirect proof to show that if –2x + 11 < 7, then x > 2.

Given: –2x + 11 < 7

Prove: x > 2

Given: ΔJKL with side lengths 5, 7, and 8 as shown.

Prove: mK < mL

Write an indirect proof.

Lesson 5.5

Inequalities Involving Two Triangles

SAS Inequality Theorem SSS Inequality Theorem Examples:

(Hinge Theorem)-When 2 sides of a triangle are congruent to 2 sides of another triangle, and the included angle of one triangle is greater than the included angle of the other triangle…

Then, the side opposite the larger angle is larger than the side opposite the smaller angle

-When 2 sides of a triangle are congruent to 2 sides of another triangle, and the 3rd side of a triangle is greater than the 3rd side of the other triangle…

Then, the angle opposite the larger side is larger than the angle opposite the smaller side

Ex:

Ex:

Ex:

On the other side of the foldable from Lesson 2 (3 column chart)

A. Compare the measures AD and BD.

B. Compare the measures ABD and BDC.

ALGEBRA Find the range of possible values for a.

Find the range of possible values of n.