Unit 3

Triangles

Chapter Objectives• Classification of Triangles by Sides

• Classification of Triangles by Angles

• Exterior Angle Theorem

• Triangle Sum Theorem

• Adjacent Sides and Angles

• Parts of Specific Triangles

• 5 Congruence Theorems for Triangles

Lesson 3.1

Classifying Triangles

Lesson 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 Sides

Name Equilateral Isosceles Scalene

Looks Like

Characteristics 3 congruent sides

At least 2 congruent

sides

No Congruent Sides

Classification of Triangles by Angles

Name Acute Equiangular Right Obtuse

Looks Like

Characteristics 3 acute angles

3 congruent angles

1 right angles

1 obtuse angle

Example 3.1Classify the following triangles by their

sides and their angles.

Scalene

Obtuse Scalene

Right

Isosceles

Acute

Equilateral

Equiangular

Vertex• 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

Adjacent Sides and Adjacent Angles• Adjacent sides are

those sides that intersect at a common vertex of a polygon.– These are said to be

adjacent to an angle.

• Adjacent angles are those angles that are right next to each other as you move inside a polygon.– These are said to be

adjacent to a specific side.

More 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

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

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

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

angles of a triangle is 180o.

A

B

C

mA + mB + mC = 180o

Example 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 3.3Solve 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

34

59

m A

m B

m C

Acute

Example 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 3.5Draw 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• The measure of an exterior angle of a

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

m A +m B = m C

A

B

C

Example 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 Theorem• 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.

C

A

B

mA + mB = 90o

Example 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 3.1• 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)

• Utilize the Triangle Inequality Theorem.

Theorem 5.10:Theorem 5.10: Side Lengths of a Triangle Side Lengths of a Triangle TheoremTheorem• If one side of a triangle is longer than

another side, then the angle opposite the longer side is larger than the angle opposite the shorter side.– Basically, the larger the side, the larger the

angle opposite that side.

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 one angle of a triangle is larger than

another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle.– Basically, the larger the angle, the larger

the side opposite that 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

Theorem 5.13: Triangle InequalityTheorem 5.13: Triangle Inequality

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

Add each combination of two sides to make Add each combination of two sides to make sure that they are longer than the third sure that they are longer than the third remaining side.remaining side.

66

66

66

113333

22 44

44

Example 12Example 12Determine whether the following could be

lengths of a triangle.

a) 6, 10, 15a) 6 + 10 > 15

10 + 15 > 66 + 15 > 10YES!

b) 11, 16, 32b) 11 + 16 < 32

NO!

Hint: A shortcut is to make sure that the sum of the two smallest Hint: A shortcut is to make sure that the sum of the two smallest sides is bigger than the third side.sides is bigger than the third side.

The other sums will always work.The other sums will always work.

Homework 3.2• Lesson 3.2 – Inequalities in One Triangle

– p7-8

• Due Tomorrow• Quiz Friday, October 15th

Lesson 3.3

Isosceles,

Equilateral,

and

Right 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)

Isosceles Triangle Theorems•Theorem 4.6: Base Angles Theorem

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

•Theorem 4.7: Converse of Base Angles Theorem

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

Example 10Solve for x

Theorem 4.7

4x + 3 = 15

4x = 12

x = 3

Theorem 4.6

7x + 5 = x + 47

6x + 5 = 47

6x = 42

x = 7

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 11

Solve for x

Corollary to Theorem 4.6

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

5x = 60

x = 12

Corollary to Theorem 4.6

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

2x + 3 = 4x - 5

3 = 2x - 5

8 = 2x

x = 4

Homework 3.3• Lesson 3.3 – Isosceles, Equilateral, and

Right Triangles– p9-11

• Due Tomorrow• Quiz Tomorrow

– Friday, October 15th

Lesson 5.3Lesson 5.3

Medians and AltitudesMedians and Altitudes

of Trianglesof Triangles

Lesson 5.3 ObjectivesLesson 5.3 Objectives• Define a median of a triangleDefine a median of a triangle

• Identify a centroid of a triangleIdentify a centroid of a triangle

• Define the altitude of a triangleDefine the altitude of a triangle

• Identify the orthocenter of a triangleIdentify the orthocenter of a triangle

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

to a segment at its to a segment at its midpointmidpoint is called the is called the perpendicular bisectorperpendicular bisector..

Triangle MediansTriangle Medians• A A median of a trianglemedian of a triangle is a segment is a segment

that does the following:that does the following:– Contains one endpoint at a Contains one endpoint at a vertexvertex of the of the

triangle, andtriangle, and– Contains the other endpoint at the Contains the other endpoint at the

midpointmidpoint of the of the opposite sideopposite side of the of the triangle.triangle.

A

B

CD

CentroidCentroid• When all three medians are drawn in, they When all three medians are drawn in, they

intersect to form the intersect to form the centroid of a trianglecentroid of a triangle..– This special This special point of concurrencypoint of concurrency is the is the

balance point for any evenly distributed balance point for any evenly distributed triangle.triangle.

• In Physics, this is how we locate theIn Physics, this is how we locate the center of mass 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:Theorem 5.7:Concurrency of Medians of a TriangleConcurrency of Medians of a Triangle• The The medians of a trianglemedians of a triangle intersect intersect

at a point that is two-thirds of the at a point that is two-thirds of the distance from each vertex to the distance from each vertex to the midpoint of the opposite side.midpoint of the opposite side.– The The centroidcentroid is is 22//33 the distance from any the distance from any

vertex to the opposite side.vertex to the opposite side.

AP = AP = 22//33AEAE

BP = BP = 22//33BFBF

CP = CP = 22//33CDCD 22 // 33AEAE

22 // 33B

FB

F

22//33 CDCD

Example 6Example 6

SS is the is the centroidcentroid of of RTW, RS = 4, VW = 6, and TV = 9. Find the following:RTW, RS = 4, VW = 6, and TV = 9. Find the following:a)a) RVRV

a)a) 66

b)b) RURUb)b) 66

• 4 is 4 is 22//33 of 6 of 6• Divide 4 by 2 and then muliply by 3. Divide 4 by 2 and then muliply by 3. Works everytime!!Works everytime!!

c)c) SUSUc)c) 22

d)d) RWRWd)d) 1212

e)e) TSTSe)e) 66

• 6 is 6 is 22//33 of 9 of 9

f)f) SVSVf)f) 33

AltitudesAltitudes• An An altitude of a trianglealtitude of a triangle is the is the perpendicularperpendicular

segment from a segment from a vertexvertex to the to the opposite sideopposite side..– It It does not does not bisectbisect the the angleangle..– It It does notdoes not bisect the bisect the sideside..

• The The altitudealtitude is often thought of as the is often thought of as the heightheight..• While true, there are While true, there are 33 altitudesaltitudes in every triangle in every triangle

but only but only 11 heightheight!!

OrthocenterOrthocenter• The three The three altitudesaltitudes of a triangleof a triangle intersect at a intersect at a

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

– inside the triangleinside the triangle– outside the triangle, oroutside the triangle, or– on one side of the triangleon one side of the triangle

AcuteAcute

RightRightObtuseObtuse

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

Theorem 5.8:Theorem 5.8:Concurrency of Altitudes of a TriangleConcurrency of Altitudes of a Triangle• The lines containing the The lines containing the altitudes of altitudes of

a trianglea triangle are are concurrentconcurrent..

Example 7Example 7Is segment BD a median, altitude, or perpendicular Is segment BD a median, altitude, or perpendicular

bisector of bisector of ABC?ABC?Hint: It could be more than one!Hint: It could be more than one!

MedianMedian

AltitudeAltitude

PerpendicularPerpendicularBisectorBisector

NoneNone

Homework 3.4• Lesson 3.4 – Altitudes and Medians

– p12-13

• Due Tomorrow

Lesson 1.7

Intro to Perimeter,

Circumference and

Area

Lesson 1.7 Objectives• Find the perimeter and area of

common plane figures.

• Establish a plan for problem solving.

Perimeter and Area of a Triangle

• The perimeter can be found by adding the three sides together.– P = a + b + c

• If the third side is unknown, use the Pythagorean Theorem to solve for the unknown side.– a2 + b2 = c2

• Where a,b are the two shortest sides and c is the longest side.

• The area of a triangle is half the length of the base times the height of the triangle.– The height of a

triangle is the perpendicular length from the base to the opposite vertex of the triangle.

– A = ½bh

ab

c

h

Homework 3.5• Lesson 3.5 – Area and Perimeter of

Triangles– p14-15

• Due Tomorrow