Unit 3

Triangles

Lesson 3.1

Classifying Triangles

Lesson 3.1 ObjectivesLesson 3.1 Objectives• Classify triangles according to their side

lengths. (G1.2.1)

• Classify triangles according to their angle measures. (G1.2.1)

• Find a missing angle using the Triangle Sum Theorem. (G1.2.2)

• Find a missing angle using the Exterior Angle Theorem. (G1.2.2)

Classification of Triangles by Classification of Triangles by SidesSides

Classification: Equilateral Isosceles Scalene

Looks Like

Characteristics3 congruent

sides2 congruent

sidesNo Congruent

Sides

Classification of Triangles by Classification of Triangles by AnglesAngles

Name Acute Equiangular Right Obtuse

Looks Like

CharacteristicsALL

acute angles

ALL congruent

angles

ONLY1 right angles

ONLY1 obtuse

angle

Example 3.1Example 3.1Classify the following triangles by their

sides and their angles.

1. Scalene

Obtuse Scalene

Right

Isosceles

Acute

Equilateral

Equiangular

2. 3.

4.

VertexVertex• The vertex of a triangle is any point at

which two sides are joined.– It is a corner of a triangle.

• There are 3 in every triangle

How to Name a TriangleHow to Name a Triangle• To name a triangle, simply draw a small

triangle followed by its vertices.– We usually try to name the vertices in

alphabetical order, when possible.• Example:

ABC

More Parts of TrianglesMore Parts of Triangles• If you were to extend the sides you will

see that more angles would be formed.• So we need to keep them separate

– There are three angles called interior angles because they are inside the triangle.

– There are three new angles called exterior angles because they lie outside the triangle.

Theorem 4.1: Triangle Sum TheoremTheorem 4.1: Triangle Sum Theorem• The sum of the measures of the interior

angles of a triangle is 180o.

mA + mB + mC = 180o

Example 3.2Example 3.2Solve for x and then classify the triangle

based on its angles.

3x + 2x + 55 = 180 Triangle Sum Theorem

5x + 55 = 180 Simplify

5x = 125 SPOE

x = 25 DPOE

Acute75o

50o

Example 3.3Example 3.3Solve for x and classify each triangle by angle measure.

1. o

o

o

( 30)

( 60)

m A x

m B x

m C x

( 30) ( 60) 180x x x 3 90 180x

3 90x 30x

o

o

o

60

30

90

m A

m B

m C

Right

2. o

o

o

(6 11)

(3 2)

(5 1)

m A x

m B x

m C x

(6 11) (3 2) (5 1) 180x x x 14 12 180x

14 168x 12x

o

o

o

83

38

59

m A

m B

m C

Acute

Example 3.4Example 3.4Draw a sketch of the triangle described.

Mark the triangle with symbols to indicate the necessary information.

1. Acute Isosceles

2. Equilateral

3. Right Scalene

Example 3.5Example 3.5Draw a sketch of the triangle described.

Mark the triangle with specific angle measures, side lengths, or symbols to indicate the necessary information.

1. Obtuse Scalene

2. Right Isosceles

3. Right Equilateral(Not Possible)

Theorem 4.2: Exterior Angle Theorem 4.2: Exterior Angle TheoremTheorem• The measure of an exterior angle of a

triangle is equal to the sum of the measures of the two nonadjacent interior angles.

1m m A m B

Example 3.6Example 3.6Solve for x

6 7 2 (103 )x x x Exterior Angles Theorem

6 7 103x x Combine Like Terms

5 7 103x Subtraction Property

5 110x Addition Property

22x Division Property

Corollary to the Triangle Sum Corollary to the Triangle Sum TheoremTheorem• A corollary to a theorem is a statement that

can be proved easily using the original theorem itself.– This is treated just like a theorem or a postulate in

proofs.

• The acute angles in a right triangle are complementary.

m A m B o90

Example 3.7Example 3.7Find the unknown angle measures.

1.

2.

3.

4.

o o o90 42 1 180m o o132 1 180m

o1 48m

o o o90 53 1 180m o o143 1 180m

o1 37m

VA

o o o90 33 2 180m o o123 2 180m

o2 57m

o o68 1 102m o1 34m

o o102 2 180m o2 78m

o o o68 34 2 180m

If you don’t like the Exterior Angle Theorem, then find m2 first using the Linear Pair Postulate.

o o102 2 180m o2 78m

Then find m1 using the Angle Sum Theorem.

o o146 1 180m o1 34m

o o o78 68 1 180m

o o58 2 180m o2 122m

VA

o2 3 122m m

o o o122 22 1 180m o o144 1 180m

o1 36m

o o o122 20 4 180m o o142 4 180m

o4 38m

Homework 3.1Homework 3.1• Lesson 3.1 – All Sections

– p1-6

• Due Tomorrow

Lesson 3.2Lesson 3.2

Inequalities in One TriangleInequalities in One Triangle

Lesson 3.2 ObjectivesLesson 3.2 Objectives• Order the angles in a triangle from

smallest to largest based on given side lengths. (G1.2.2)

• Order the side lengths of a triangle from smallest to largest based on given angle measures. (G1.2.2)

Theorem 5.10:Theorem 5.10: Side Lengths of a Triangle Side Lengths of a Triangle TheoremTheorem• If two sides of a triangle unequal, then the measures

of the angles opposite theses sides are also unequal, with the greater angle being opposite the greater side.– Basically, the largest angle is found opposite the largest

side.– Basically, the largest side is found opposite the largest

angle.

Longest side

Longest side

Largest AngleLargest Angle 22ndnd Longest Side Longest Side

22ndnd Largest Largest AngleAngle

Smallest Smallest SideSide

Smallest Smallest AngleAngle

Theorem 5.11: Angle Measures of a Triangle Theorem 5.11: Angle Measures of a Triangle TheoremTheorem• If two angles of a triangle unequal, then the

measures of the sides opposite theses angles are also unequal, with the greater side being opposite the greater angle.– Basically, the largest angle is found opposite the largest

side.– Basically, the largest side is found opposite the largest

angle.

Longest side

Longest side

Largest AngleLargest Angle 22ndnd Longest Side Longest Side

22ndnd Largest Largest AngleAngle

Smallest Smallest SideSide

Smallest Smallest AngleAngle

Example 3.8Example 3.8Order the angles from largest to smallest.

1. , ,B A C 2. , ,Q P R

3. , ,A C B

Example 3.9Example 3.9Order the sides from largest to smallest.1.

2.

, ,ST RS RT

, ,DE EF DF

33o

Example 3.10Order the angles from largest to smallest.

1. In ABCAB = 12BC = 11AC = 5.8

Order the sides from largest to smallest.

2. In XYZmX = 25o

mY = 33o

mZ = 122o

, ,C A B

, ,XY XZ YZ

Homework 3.2• Lesson 3.2 – Inequalities in One Triangle

– p7-8

• Due Tomorrow• Quiz Friday, October 15th

Lesson 3.3

Isosceles and

Equilateral Triangles

Lesson 3.3 Objectives• Utilize the Base Angles Theorem to

solve for angle measures. (G1.2.2)

• Utilize the Converse of the Base Angles Theorem to solve for side lengths. (G1.2.2)

• Identify properties of equilateral triangles to solve for side lengths and angle measures. (G1.2.2)

Special Parts of an Isosceles Triangle

• An isosceles triangle has only two congruent sides– Those two congruent sides are called

legs.– The third side is called the base.

legs

base

Isosceles Triangle Theorems•Theorem 4.6: Base Angles Theorem

–If two sides of a triangle are congruent, then the angles opposite them are congruent to each other.

•Theorem 4.7: Converse of Base Angles Theorem

–If two angles of a triangle are congruent, then the sides opposite them are congruent.

Example 3.11Solve for x and y.1.

2.

3.

4.

7x

o75x

75o

75 75 180x 150 180x

30x

3 11 2 11x x 11 11x

22x

2(22) 11 44 11 55

55o

55o

55 55 2 180y 110 2 180y

2 70y 35y

5. = 90o

= 90o

+45o 45o= 4545 =

3 45x 15x

7 45y 38y

Equilateral Triangles•Corollary to Theorem 4.6

–If a triangle is equilateral, then it is equiangular.

•Corollary to Theorem 4.7–If a triangle is equiangular, then it is equilateral.

Example 3.12Solve for x and y.1.

Or…In order for a triangle to be equiangular, all angles must equal…

It does not matter which two sides you set equal to each other, just pick the pair that looks the easiest!

5xo 5xo

5 5 5 180x x x 15 180x

12x

2.

5 60x 12x

2 3 4 5x x 3 2 5x 8 2x

4x

Homework 3.3• Lesson 3.3 – Isosceles and Equilateral Triangles

– p9-11

• Due Tomorrow• Quiz Tomorrow

– Tuesday, October 19th

Lesson 3.4Lesson 3.4

MediansMedians

AndAnd

Altitudes of TrianglesAltitudes of Triangles

Lesson 3.4 ObjectivesLesson 3.4 Objectives• Identify a median, an altitude, and a

perpendicular bisector of a triangle. (G1.2.5)

• Identify a centroid of a triangle.• Utilize medians and altitudes to solve

for missing parts of a triangle. (G1.2.5)

• Identify the orthocenter of a triangle.

Perpendicular BisectorPerpendicular Bisector• A segment, ray, line, or plane that is perpendicular to

a segment at its midpoint is called the perpendicular bisector.

Triangle MediansTriangle Medians• A median of a triangle is a segment that does the

following:– Contains one endpoint at a vertex of the triangle,

and– Contains its other endpoint at the midpoint of the

opposite side of the triangle.

A

B

CD

CentroidCentroid• When all three medians are drawn in, they

intersect to form the centroid of a triangle.– This forms a point of concurrency which is defined as a

point formed by the intersection of two or more lines.

• The centroid happens to find the balance point for any triangle.

• In Physics, this is how we locate the center of mass.

AcuteAcute RightRight

ObtuseObtuse

Remember: All Remember: All medians medians intersect the intersect the midpointmidpoint of the opposite side.of the opposite side.

Theorem 5.7: Concurrency of Medians of a Theorem 5.7: Concurrency of Medians of a TriangleTriangle• 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.– The centroid is 2/3 the distance from any vertex to the

opposite side.• Or said another way, the centroid is twice as far away from the

opposite angle as it is to the nearest side.

AP = AP = 22//33AEAE

BP = BP = 22//33BFBF

CP = CP = 22//33CDCD

Example 3.13Example 3.13

S is the centroid of RTW, RS = 4, VW = 6, and TV = 9. Find the following:a) RV

a) 6

b) SUb) 2

• Half of 4 is 2

c) RUc) 6

• 4 + 2 = 6

d) RWd) 12

e) TSe) 6

• 6 is 2/3 of 9

f) SVf) 3

• Half of 6, which is the other part of the median.

AltitudesAltitudes• An altitude of a triangle is the perpendicular

segment from a vertex to the opposite side.– It does not bisect the angle.– It does not bisect the side.

• The altitude is often thought of as the height.– While true, there are 3 altitudes in every triangle but only 1

height!

OrthocenterOrthocenter• The three altitudes of a triangle intersect at a point

that we call the orthocenter of the triangle.• The orthocenter can be located:

– inside the triangle – outside the triangle, or– on one side of the triangle

AcuteAcute

RightRight

ObtuseObtuse

The The orthocenterorthocenter of a right triangle of a right triangle will always be located at the vertex will always be located at the vertex that forms the right angle.that forms the right angle.

Example 3.14Example 3.14Is segment BD a median, altitude, or perpendicular bisector of ABC?Hint: It could be more than one!1.

MedianMedianAltitudeAltitude

PerpendicularPerpendicularBisectorBisector

NoneNone

2.

3.

4.

MedianMedian

NoneNone

Homework 3.4• Lesson 3.4 – Altitudes and Medians

– p12-13

• Due Tomorrow

Lesson 3.5

Area

and

Perimeter of Triangles

Lesson 3.5 Objectives• Find the perimeter and area of

triangles. (G1.2.2)

Reviewing AltitudesDetermine the size of the altitudes of the following

triangles.I.

6

II. 16

III. ?

If it is a right triangle, then you can use Pythagorean Theorem to solve for the missing side length.

a

b

c

2 2 2a b c 2 2 26 10a 2 36 100a

2 64a 64a 8

Area• The area of a figure is defined as “the amount of space inside the boundary of a flat (2-

dimensional) object”– http://www.mathsisfun.com/definitions/area.html

• Because of the 2-dimensional nature, the units to measure area will always be “squared.”– For example:

• in2 or square inches

• ft2 or square feet

• m2 or square meters

• mi2 or square miles

• The area of a rectangle has up until now been found by taking:• length x width (l x w)

• We will now change the wording slightly to fit a more general pattern for all shapes, and that is:• base x height (b x h)

• That general pattern will exist as long asthe base and height form a right angle.

– Or said another way, the base andheight both touch the right angle.

w

l

h

b

Area of a Triangle• The area of a triangle is found by taking one-half

the base times the height of the triangle.• Again the base and height form a right angle.

– Notice that the base is an actual side of the triangle, and…

– The height is nothing more than the altitude of the triangle drawn from the base to the opposite vertex.

h

b

h

b

( )

1

2A b h

Perimeter of a Triangle• The perimeter of a triangle is found by taking the sum of all three

sides of the triangle.– So basically you need to add all three sides together.

• The perimeter is a 1-dimensional measurement, so the units should not have an exponent on them.

– Example:» in» ft» m» mi

c

( )P a b c

hab

Example 3.15Find the area and perimeter of the following triangles.

1.

( )

1

2A b h

( )

1(24)(10)

2A

( )(12)(10)A

( )120 sq. unitsA

( )P a b c

( )10 24 26P

( )60 unitsP

2.

( )

1

2A b h

( )

1(10.5)(6)

2A

( )(10.5)(3)A

( )31.5 sq. unitsA

( )P a b c

( )10 6.5 10.5P

( )27 unitsP

Homework 3.5• Lesson 3.5 – Area and Perimeter of

Triangles– p14-15

• Due Tomorrow