Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

Triangles and their properties

•Triangle Angle sum Theorem•External Angle property•Inequalities within a triangle•Triangle inequality theorem•Medians•Altitude•Perpendicular Bisector•Angle Bisector

2

Triangle Angle Sum Theorem• The sum of the measures of the angles of a

triangle is 180°. m∠A + m∠B + m∠C = 180

A

B

C

Ex: If m∠A = 30 and m∠B = 70; what is m∠C ?

m∠A + m∠B + m∠C = 180 30 + 70 + m∠C = 180 100 + m∠C = 180 m∠C = 180 – 100 = 80

Exterior Angle Theorem

1

2 3 4

P

Q RIn the triangle below, recall that 1, 2, and 3 are _______ angles of ΔPQR.interior

Angle 4 is called an _______ angle of ΔPQR.exterior

An exterior angle of a triangle is an angle that forms a _________, (they add up to 180) with one of the angles of the triangle.linear pair

____________________ of a triangle are the two angles that do not forma linear pair with the exterior angle.

Remote interior angles

In ΔPQR, 1, and 2 are the remote interior angles with respect to 4.

In ΔPQR, 4 is an exterior angle because 3 + 4 = 180 .

The measure of an exterior angle of a triangle is equal to sum

of its ___________________remote interior angles

Exterior Angle Theorem

1

2

3 4 5

In the figure, which angle is the exterior angle? 5

which angles are the remote the interior angles? 2 and 3

If 2 = 20 and 3 = 65 , find 5

65

20

If 5 = 90 and 3 = 60 , find 2

85

90 60

30

Exterior Angle Theorem

Exterior Angle Theorem

3 and 1

Inequalities Within a Triangle

If the measures of three sides of a triangle are unequal,

then the measures of the angles opposite those sides

are unequal ________________.

13

811

L

P

M

in the same order

LP < PM < ML

mM < mPmL <

Inequalities Within a Triangle

If the measures of three angles of a triangle are unequal,

then the measures of the sides opposite those angles

are unequal ________________.in the same order

JK < KW < WJ

mW < mKmJ <

J

45°W K

60°

75°

Inequalities Within a Triangle Inequalities Within a Triangle

In a right triangle, the hypotenuse is the side with the

________________.greatest measure

WY > XW

35

4 Y

W

X

WY > XY

Inequalities Within a Triangle

A

The longest side is BC

So, the largest angle is

LThe largest angle is

MNSo, the longest side is

Triangle Inequality Theorem

TriangleInequalityTheorem

The sum of the measures of any two sides of a triangle is

_______ than the measure of the third side.greater

a

b

c

a + b > c

a + c > b

b + c > a

Triangle Inequality Theorem

Can 16, 10, and 5 be the measures of the sides of a triangle?

No! 16 + 10 > 5

16 + 5 > 10

However, 10 + 5 > 16

Medians, Altitudes, Angle Bisectors

Perpendicular Bisectors

Every triangle has 1. 3 medians, 2. 3 angle bisectors and 3. 3 altitudes.

A

B

C

Given ABC, identify the opposite side

1. of A.

2. of B.

3. of C.

BC

AC

AB

Just to make sure we are clear about what an opposite side is…..

A new term…

Point of concurrency

• Where 3 or more lines intersect

Any triangle has three medians.

B

A

C

M

N

L

Let L, M and N be the midpoints of AB, BC and AC respectively. CL, AM and NB are medians of ABC.

Definition of a Median of a Triangle

A median of a triangle is a segment whose endpoints are a vertex and a midpoint of the

opposite side

The point where all 3 medians intersectThe point where all 3 medians intersect

CentroidCentroidIs the point of Is the point of concurrencyconcurrency

The The centroidcentroid is 2/3’s of the distance is 2/3’s of the distance from the vertex to the side.from the vertex to the side.

2x2x

xx

1010

55

3232

XX1616

The The centroidcentroid is the center of balance is the center of balance for the triangle. You canfor the triangle. You can

balance a triangle on the tip ofbalance a triangle on the tip ofyour pencil if you place the tip onyour pencil if you place the tip on

the the centroidcentroid

angle bisector of a triangle

a segment that bisects an angle of the triangle and goes to the opposite side.

A

B

CD

E F

In the figure, AF, DB and EC

are angle bisectors of ABC.

Any triangle has three angle bisectors.

Note: An angle bisector and a median of a triangle are sometimes different.

M

Let M be the midpoint of AC. The median goes from the vertex to the midpoint of the opposite side.

BM is a medianBD is a angle bisector of ABC.

The Incenter is where all The Incenter is where all 3 Angle bisectors intersect3 Angle bisectors intersect

Incenter Incenter Is the point of concurencyIs the point of concurency

Any point on an angle bisector is Any point on an angle bisector is equidistance from both sides of the angle equidistance from both sides of the angle

This makes the This makes the IncenterIncenter an anequidistance from all 3 sidesequidistance from all 3 sides

D

Theorem: If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle.

Let AD be a bisector of BAC,

P lie on AD,

PM AB at M,

NP AC at N.

A

B

C

M

N

P

Then P is equidistant from AB and AC.

Theorem:Theorem: If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle.

The converse of this theorem is not always true.

Theorem:Theorem: If a point is in the interior of an angle and is equidistant from the sides of the angle, then the point lies on the bisector of the angle.

Using the Angle Bisector Theorem• What is the length of RM?Because angle N has been bisected, I know that each point along the bisector is equidistant to the sides

Since MR and RP are both perpendicular to each side and touch the bisector, I know they are equal

7x = 2x + 25

5x = 25

x = 5

What is the length of FB?Because angle C has been bisected, I know that each point along the bisector is equidistant to the sides

Since BF and FD are both perpendicular to each side and touch the bisector, I know they are equal

6x +3 = 4x + 9

2x +3 = 9

2x = 6

x = 3

Any triangle has three altitudes.

Definition of an Altitude of a Triangle

A altitude of a triangle is a segment that has one endpoint at a vertex and the other creates a right angle at the opposite side.

The altitude is perpendicular to the opposite side while going through the vertex

ACUTE OBTUSE

B

A

C

RIGHT

A

B C

If ABC is a right triangle, identify its altitudes.

BG, AB and BC are its altitudes.

G

Can a side of a triangle be its altitude?YES!

Orthocenter is where all the Orthocenter is where all the altitudes intersect.altitudes intersect.

OrthocenterOrthocenter

The The orthocenterorthocenter can be located can be locatedin the triangle, on the triangle orin the triangle, on the triangle oroutside the triangle.outside the triangle.

RightRight

Legs are altitudesLegs are altitudes

ObtuseObtuse

A Perpendicular bisector of a side does A Perpendicular bisector of a side does not have to start at a vertex. It will formnot have to start at a vertex. It will form

a a 90° angles90° angles and bisectand bisect the side. the side.

CircumcenterCircumcenterIs the point of concurrencyIs the point of concurrency

Any point on the Any point on the perpendicular bisectorperpendicular bisectorof a segment is equidistance from theof a segment is equidistance from the

endpoints of the segment.endpoints of the segment.

AA

BB

CC DD

AB is the perpendicularAB is the perpendicularbisector of CDbisector of CD

This makes the This makes the CircumcenterCircumcenter an anequidistance from the 3 verticesequidistance from the 3 vertices

Perpendicular Bisector

Perpendicular Bisector

Using the Perpendicular Bisector Theorem

• What is the length of AB?

Since BD is a perpendicular bisector, I know that BA and BC are congruent since they are connected to the vertex and the end of the bisected line.

4x = 6x – 10

–2x = – 10

x = 5

Since BD perpendicular to the side opposite B and bisects AC,I know that BD is a perpendicularbisector.

AB = 4x AB = 4(5) AB = 20

BC = 6x – 10

BC = 6(5) – 10

BC = 20

Since SQ is perpendicular to the side opposite Q and bisects PR,I know that SQ is a perpendicularbisector.

What is the length of QR?

3n – 1= 5n – 7Since SQ is a perpendicular bisector, I know that PQ and QR are congruent since they are connected to the vertex and the end of the bisected line.

– 1= 2n – 76 = 2n 3 = n

PQ = 3n – 1

PQ = 3(3) –1

PQ = 8

QR = 5(3) – 7

QR = 5(n) – 7

QR = 8

The Midsegment of a Triangle is a segment that connects the midpoints of

two sides of the triangle.

D

B

C

E

A

D and E are midpoints

DE is the midsegment

The midsegment of a triangle is parallel to the third side and is half as long as that side.

DE AC1

DE AC2

The midsegment of a triangle is parallel to the third side and is half as long as that side.

1DE AC

2

DE AC

Midsegment Theorem

D

B

C

E

A

1. Identify the 3 pairs of parallel lines shown above

UW TX

WY VT

YU XV

2a. = 46,

what is ?

If LK

NM

2b.

MN is half as long as LK

2(MN) = 46

JK = 5x + 20 and

NO = 20, find x

If

NO is half and big as JK

2(20) = 5x +20

40 = 5x + 20

MN = 23 x = 4

Example 1In the diagram, ST and TU are midsegments of

triangle PQR. Find PR and TU.

PR = ________ TU = ________16 ft 5 ft

Example 2In the diagram, XZ and ZY are midsegments of

triangle LMN. Find MN and ZY.

MN = ________ ZY = ________53 cm 14 cm

Example 3In the diagram, ED and DF are midsegments of

triangle ABC. Find DF and AB.

DF = ________ AB = ________26 52

3X – 4

5X+2

x = ________10

2 (DF ) = AB

2 (3x – 4 ) = 5x + 2

6x – 8 = 5x + 2

x – 8 = 2

x = 10

Perpendicular Bisectors

• A point is equidistant from two objects if it is the same distance from each.

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

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

Angle Bisectors

• The distance from a point to a line is the length of the perpendicular segment from the point to the line.

Angle Bisector Theorem: If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle.

Converse of the Angle Bisector Theorem: If a point in the interior of an angle is equidistant from the sides of the angle, then the point is on the angle bisector.

There are 3 of each of these specialThere are 3 of each of these special segments in a triangle.segments in a triangle.

The 3 segments are concurrent. TheyThe 3 segments are concurrent. Theyintersect at the same point.intersect at the same point.

This point is called the point of This point is called the point of concurrency.concurrency.

The points have special names and The points have special names and special properties.special properties.

Altitude ..Altitude .. Vertex .. 90° .. Vertex .. 90° .. OrthocenterOrthocenter

Angle BisectorAngle Bisector.... Angle into 2 equal angles .. Angle into 2 equal angles .. IncenterIncenter

Perpendicular BisectorPerpendicular Bisector…… 90° .. bisects side .. 90° .. bisects side .. CircumcenterCircumcenter

MedianMedian .. .. Vertex .. Midpoint of side ..Vertex .. Midpoint of side ..CentroidCentroid

Give the best name for ABGive the best name for ABAA

BB

AA

BB

AA

BB

AA

BB

AA

BB||||

|| ||

||||

MedianMedian AltitudeAltitude NoneNone AngleAngleBisectorBisector

PerpendicularPerpendicularBisectorBisector

Survival TrainingSurvival TrainingYou’re Stranded On A Triangular You’re Stranded On A Triangular

Shaped Island. The Rescue Ship CanShaped Island. The Rescue Ship CanOnly Dock On One Side Of The Island Only Dock On One Side Of The Island

But You Don’t Know Which Side. At But You Don’t Know Which Side. At Which Point Of Concurrency Would Which Point Of Concurrency Would

You Set Up Camp So You Are An You Set Up Camp So You Are An EqualEqual Distance From All 3 Sides?Distance From All 3 Sides?

INCENTERINCENTER

What If The Ship Could OnlyWhat If The Ship Could OnlyDock At One Of The Vertices? Dock At One Of The Vertices? Would You Change The Would You Change The Location Of Your Camp ?Location Of Your Camp ?If So, Where?If So, Where?

YESYES CIRCUMCENTERCIRCUMCENTER

Where would you place a fire hydrant toWhere would you place a fire hydrant tomake it equidistance to the houses andmake it equidistance to the houses and

equidistance to the streets?equidistance to the streets?

ELM

ELM

POSTPOST

ELM

ELM

POSTPOST

Angle bisector for the streetsAngle bisector for the streetsPerpendicular bisector for housesPerpendicular bisector for houses