Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities...

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

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Transcript of Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities...

Page 1: 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

Page 2: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 3: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 4: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 5: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

Exterior Angle Theorem

Page 6: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

Exterior Angle Theorem

3 and 1

Page 7: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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 <

Page 8: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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°

Page 9: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 10: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

Inequalities Within a Triangle

A

The longest side is BC

So, the largest angle is

LThe largest angle is

MNSo, the longest side is

Page 11: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 12: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 13: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

Medians, Altitudes, Angle Bisectors

Perpendicular Bisectors

Page 14: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 15: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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…..

Page 16: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

A new term…

Point of concurrency

• Where 3 or more lines intersect

Page 17: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 18: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

CentroidCentroidIs the point of Is the point of concurrencyconcurrency

Page 19: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 20: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 21: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

angle bisector of a triangle

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

Page 22: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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.

Page 23: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 24: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 25: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 26: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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.

Page 27: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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.

Page 28: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 29: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 30: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 31: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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!

Page 32: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

OrthocenterOrthocenter

Page 33: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 34: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 35: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 36: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 37: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

Perpendicular Bisector

Page 38: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

Perpendicular Bisector

Page 39: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 40: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 41: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 42: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 43: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

1. Identify the 3 pairs of parallel lines shown above

UW TX

WY VT

YU XV

Page 44: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 45: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

triangle PQR. Find PR and TU.

PR = ________ TU = ________16 ft 5 ft

Page 46: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

triangle LMN. Find MN and ZY.

MN = ________ ZY = ________53 cm 14 cm

Page 47: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 48: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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.

Page 49: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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.

Page 50: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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.

Page 51: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 52: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 53: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 54: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 55: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

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

Page 56: Triangles and their properties Triangle Angle sum Theorem External Angle property Inequalities within a triangle Triangle inequality theorem Medians Altitude.

ELM

ELM

POSTPOST

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


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