Unit 3

Triangles and their propertiesTriangles and their properties

Lesson 3.1

Classifying TrianglesClassifying Triangles

Triangle Sum TheoremTriangle Sum Theorem

Exterior Angle TheoremExterior Angle Theorem

Classification of Triangles by Sides

NameName EquilateralEquilateral IsoscelesIsosceles ScaleneScalene

Looks LikeLooks Like

CharacteristicsCharacteristics 33 congruent congruent sidessides

At At least 2least 2 congruent sidescongruent sides

NoNo Congruent Congruent SidesSides

Classification of Triangles by Angles

NameName AcuteAcute EquiangularEquiangular RightRight ObtuseObtuse

Looks LikeLooks Like

CharacteristicsCharacteristics 33 acuteacute anglesangles

33 congruentcongruent anglesangles

11 rightright anglesangles

11 obtuseobtuse angleangle

Example 1

You must classify the triangle as specific as You must classify the triangle as specific as you possibly can.you possibly can.

That means you must nameThat means you must name Classification according to anglesClassification according to angles Classification according to sidesClassification according to sides

In that order!In that order! ExampleExample

Obtuse isosceles

More Examples

8.6 8.6

8.6

600

600

600

2.5 2.5

4.5

260 260

1280

Equiangular

Equilateral

Obtuse

Isosceles

And more examples…Draw a sketch of the following triangles. Use proper symbol notation

Obtuse Scalene Equilateral Right Equilateral

impossible

Proving the Sum of a triangle’s Angles

What do we know about the two green angles labeled X?

What do we know about the two yellow angles labeled Y?

What do we know about the three angles at the top of the triangle? (the X, Y and Z)

Proving the Sum of a triangle’s Angles

X=X because of Alternate interior angles.

What do we know about the two yellow angles labeled Y?

What do we know about the three angles at the top of the triangle? (the X, Y and Z)

Proving the Sum of a triangle’s Angles

X=X because of Alternate interior angles.

Y=Y because of Alternate interior angles.

What do we know about the three angles at the top of the triangle? (the X, Y and Z)

Proving the Sum of a triangle’s Angles

X=X because of Alternate interior angles.

Y=Y because of Alternate interior angles.

X+Z+Y=180 because they form a straight line.

Proving the Sum of a triangle’s Angles

X=X because of Alternate interior angles.

Y=Y because of Alternate interior angles.

X+Z+Y=180 because they form a straight line.

The sum of the interior angles must also be equal to 1800

Triangle Sum Theorem

The sum of the interior angles of a triangle The sum of the interior angles of a triangle is 180.is 180.

Let’s try some…

248 m m 1=37

2 57m

Proving the Exterior Angle Theorem

180m a m b c triangle sumtheorem

180m b m d Linear Pair a+b+c=b+d Substitution

a+c=d Subtraction (subtract b from both sides)

Exterior Angle Theorem

Practice(7x+1)+38=10x+9

7x+39=10x+9

39=3x+9

30=3x

10=x

Lesson 3.2

Inequalities in one triangleInequalities in one triangle

Side/Angle Pairs in a Triangle

A

B

C

Angle A and the side opposite it are a pair

Angle B and the side opposite it are a pair

Angle C and the side opposite it are a pair

The Inequalities in One Triangle

If it is the longest side, then it is opposite If it is the longest side, then it is opposite largest angle measure.largest angle measure.

If it is the shortest side, then it is opposite If it is the shortest side, then it is opposite the smallest angle measure.the smallest angle measure.

If it is the middle length side, then it is If it is the middle length side, then it is opposite the middle angle measure.opposite the middle angle measure.

and their converse, too!

If it is the largest angle measure, then it is If it is the largest angle measure, then it is opposite the longest side.opposite the longest side.

If it is the smallest angle measure, then it is If it is the smallest angle measure, then it is opposite the shortest side.opposite the shortest side.

If it is the middle angle measure, then it is If it is the middle angle measure, then it is opposite the middle length side.opposite the middle length side.

Examples: Order the angles from smallest to largest.

L

M

K

B

D

C

More examples…Order the angles from smallest to largest

12

11

5.8

In UVW

VW

UW

UV

U

V

W

5.8 12

11

W

V

U

Let’s practice the converse now! Order the sides from shortest to longest.

FD

FE and ED

AC

AB

CB

Week’s Schedule

Mon: Lesson 3.3Mon: Lesson 3.3 Tue: Lesson 3.4Tue: Lesson 3.4 Wed: MEAPWed: MEAP Thu: Quiz/Lesson 3.5Thu: Quiz/Lesson 3.5 Fri: Practice testFri: Practice test

Mon: Review Unit 3Mon: Review Unit 3 Tue: Unit 3 TestTue: Unit 3 Test

Lesson 3.3

Base Angles Theorem and its ConverseBase Angles Theorem and its Converse

Let’s talk; What do we know about the following triangles?

Base Angles Theorem and Converse

Examples: solve for x and/or y

7

75

30

If the two angles are equal and the interior angles of a triangle have a sum of 180, what is the measure of the two angles?

3x = 45

x = 15

y+7 = 45

y = 38

One more example…solve for x and y

3x-11 = 2x+11x –11 = 11

x = 22

Using the value of x, the measure of the two angles are each… 55 degrees

Using the triangle sum theorem the last angle measure is… 70 degrees

2y = 70

y = 35

Corollaries (a statement that is easily proven using the original theorem)

Prove what the angles in an equilateral triangle MUST always be.

If all the sides are the same, equilateral, then all the angles must be the same, equiangular.

If one of the angles is x, then all of the angles must also be x.

x + x + x = 180 (triangle sum)3x = 180 (combine like terms)x = 60 (DPOE)

Examples

y

5x = 60x = 12

2x – 3 = 7

2x = 10x = 5

One more!

All sides are equal. Pick any two and set them equal to each other. Then solve for x.

12x – 13 = 2x + 17

10x –13 = 17

10x = 30

x = 3

Lesson 3.4

Altitudes, Medians, and Perpendicular Altitudes, Medians, and Perpendicular Bisectors of TrianglesBisectors of Triangles

Putting old terms together…

Perpendicular:Perpendicular:

Two lines that intersect at a right angle.Two lines that intersect at a right angle. Bisector:Bisector:

A segment, ray, or line that divides a segment into A segment, ray, or line that divides a segment into two congruent parts.two congruent parts.

Perpendicular bisector (of a triangle):Perpendicular bisector (of a triangle):

A segment, ray, or line that is perpendicular to a A segment, ray, or line that is perpendicular to a side of a triangle at the midpoint of the side.side of a triangle at the midpoint of the side.

Is segment BD a perpendicular bisector? Explain!

No, it is nether perp. nor a bisector.

Yes, it is perp. to segment AC and divides it into two congruent parts.

No, it is a bisector but is not perp.

No, segment BD is perp. to segment BC, but is not its bisector

New vocabulary terms!Median of a TriangleMedian of a Triangle: A segment whose : A segment whose endpoints are a vertex of the triangle and the endpoints are a vertex of the triangle and the midpoint of the opposite side.midpoint of the opposite side.

Altitude of a TriangleAltitude of a Triangle: The perpendicular : The perpendicular segment from a vertex to the opposite side or segment from a vertex to the opposite side or to the line that contains the opposite sideto the line that contains the opposite side

Is segment BD a Median? Altitude? Explain!

Neither, it is not a bisector and it is not perp.

Both, it is a bisector and is perp.

Median, it is a bisector of segment AC

DAltitude, it is perp. to segment BC

Special notes about Perp. Bisectors and Medians: All perp. bisectors are also medians.All perp. bisectors are also medians. Some medians are perp. bisectors.Some medians are perp. bisectors. If it’s not a median, then it is not a perp. If it’s not a median, then it is not a perp.

bisector.bisector.

Special notes about Perp. Bisectors and Altitudes: All perp. bisectors are also altitudes.All perp. bisectors are also altitudes. Some altitudes are perp. bisectors.Some altitudes are perp. bisectors. If it’s not a altitude, then it is not a perp. If it’s not a altitude, then it is not a perp.

bisector.bisector.

Lesson 3.5

Perimeter and Area of TrianglesPerimeter and Area of Triangles

Review: Identify the altitude of the following triangles.

Area Formula of a Triangle

1

2A b h

b is the base

h is the height

HINT: The base and the height always meet at a right angle

Find the area of the triangles

1(24)(10)

2A

12(10)A

120A

1(10.5)(6)

2A

3(10.5)A

31.5A

Find the area.

8

1(36)(8)

2A

18(8)A 144A

Formula for the perimeter of a triangle.P=a+b+cP=a+b+c

a, b, and c are the three sides of the triangle.a, b, and c are the three sides of the triangle.

HINT: Perimeter is the sum of all three sides HINT: Perimeter is the sum of all three sides of a trianlgeof a trianlge

Find the perimeter of the triangles

P=26+24+10

P=60P=6.5+10+10.5

P=27

Find the perimeter

31 P=31+10+36

P=77