Unit: Triangles

3-4 Parallel lines and Triangle Sum Theorem

Objective: To classify triangle and find the measure of their

anglesTo use exterior angle Theorem

Classifying Triangle

Classify by angles:

Classify by sides:

TheoremsTriangle-Angle Sum Theorem: Sum of the angles

is 180.

Exterior Angle Theorem: sum of the remote interior angles equals the exterior angle

C + B = BAD

1 2

Exterior angle

Remote interior angles

Example 1Using the Exterior Angle Theorem

63+56 = 119X = 119

Example 2: Exterior angle & Sum of the angle of a triangle

90-55 = 35

180-(86+35) = 5986-55 = 31

TRY

Find the measure of each angle.

180-(56+62) =62

62-25 = 37

180 – 62 = 118OR56 + 62 = 118

Example 3

Classify the triangles.a) By its sides 18cm, 20 cm, 18cmisoscelesb) By its angles 91,20 ,69 obtuse

Try

Classify the triangle. The measure of each angle is 60.

Equilateral and equiangular

Closure

1) What is the sum of the interior angles of a triangle?

180

2) What is the relationship of the exterior-angle and the two remote interior angles?

Sum of the remote interior angles = exterior angle.

3-5 Polygon Angle Sum

Objective:To classify polygonsTo find the sums of the measures of the interior

and exterior angles of a polygon

Vocabulary

• Polygon– closed figure with the at least three segments.

• Concave Convex• Equilateral polygon– All sides congruent

• Equiangular polygon – All angles congruent

• Regular polygon– Equilateral and equiangular polygon

concave

concave

convexconvex

convex

Polygon NamesName Number

of sidesSum of the interior angles

An interior angles

An exterior angles of a regular polygon

Triangle 3 60 60 120

Quadrilateral 4 360 90 90

Pentagon 5 540 108 72

Hexagon 6 720 120 60

Heptagon 7 900 900/7 51 3 /7

Octagon 8 1080 135 45

Nonagon 9 1260 140 40

Decagon 10 1440 144 36

Unagon 11 1620 1620/11 32 /11

Dodecagon 12 1800 150 30

N-gon n (n-2)180 (n-2)180 n

360/n

Polygons

Polygon Angle Sum Theorem180(n-2)

Polygon Exterior Angle Theorem:Sum of all exterior angles is 360 degrees.

Example 1

• Name the polygon by its sides• Concave or convex.• Name the polygon by its vertices. • Find the measure of the missing angle

Pentagon

Convex

QRSTU

(5-2)180 = 540

130+54+97+130 = 411540 – 411 = 129

Example 2

Find the measure of an interior and an exterior angle of the regular polygon..

360/7 = 51 3/7

(7-2)180/7 = 128 4/7

Determine the number of Sides

• If the sum of the interior angles of a regular polygon is 1440 degrees.

1440 = 180(n-2) 8 = n-2 10 = n it is a decagon • Find the measure of an exterior angle360/10 = 36 degrees

Closure

• What is the formula to find the sum of the interior angles of a polygon?

(n-2)180• What is the name of the polygon with 6 sides?hexagon• How do you find the measure of an exterior

angle?Divide the 360 by the number of sides.

4-5 Isosceles and Equilateral Triangles

Objective:To use and apply properties of isosceles and

equilateral triangles

Isosceles Triangle Key Concepts

• Isosceles Triangle Theorem

• Converse of the Isosceles Triangle

• Theorem

Isosceles Triangle Key Concepts• If a segment, ray or line bisects the vertex

angle, then it is the perpendicular bisector of the base.

Equilateral Triangle Key Concepts

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

• If the triangle is equiangular, then it is equilateral.

What did you learn today?

• What is still confusing?

5-1 Midsegments

Objective: To use properties of midsegments to solve

problems

Key Concept

Midsegments – DE = ½AB and DE || AB

Try 1

Find the perimeter of ∆ABC. 16+12+14 = 42

Try 2:

a) If mADE = 57, what is the mABC?57°b) If DE = 2x and BC = 3x +8, what is length of

DE? 4x = 3x+8 x = 8 DE = 2(8) = 16

What have you learned today?

What is still confusing?

7-1 Ratios and Proportions

Objective: To write ratios and solve proportions.

VOCBULARY

• RATIO- COMPARISON OF TWO QUANTITIES.• PROPORTION- TWO RATIOS ARE EQUAL.• EXTENDED PROPORTION – THREE OR MORE

EQUILVANT RATIOS.

PROPERTIES OF RATIOS

a c is equivalent to: 1) ad = bcb d 2) b d 3) a b a c c d

4) a + b c + d b c

Example 1

a) 5 20 x 3

b) 18 6 n + 6 n

• 15 = 20x• ¾ = x

• 18n = 6n +36• 12n = 36• n = 3

Example 2

• 1 7/8 16 x

• X = 16 (7/8)• X = 14 ft

The picture above has scale 1in = 16ft to the actual water fallIf the width of the picture is 7/8 inches, what is the size of the actual width of the part of the waterfall shown.

7-2 Similar Polygons

Objective: to identify and apply similar polygons

Vocabulary

• Similiar polygons- (1) corresponding angles are congruent and (2) corresponding sides are proportional. ( ~)

• Similarity ratio – ratio of lengths of corresponding sides

Example 1

• Find the value of x, y, and the measure of angle P.

• <P = 86• 4/6 = 7/Y X/9 = 4/6• 4Y = 42 6X = 36• Y = 10.5 X = 6

Example 2

Find PT and PR 4 = X

11 X+12 11X = 4X + 48 7X = 48 X = 6 PT = 6 PR = 18

Example 3

Hakan is standing next to a building whose shadow is 15 feet long. If Hakan is 6 feet tall and is casting a shadow 2.5 feet long, how high is the building?

X = 156 2.5 2.5X = 80X =

TRY

• A vertical flagpole casts a shadow 12 feet long at the same time that a nearby vertical post 8 feet casts a shadow 3 feet long. Find the height of the flagpole. Explain your answer.

5-2 Bisectors in Triangles

objective: To use properties of perpendicular bisectors and

angle bisectors

Key Concept

Perpendicular bisectors – forms right angles at the base(side) and bisects the base(side).

Angle Bisectors– bisects the angle and equidistant to the side.

Try 1

WY is the bisector of XZ 4 7.5 9 Isosceles triangle

Try 2

6y = 8y -7 7 = 2y y = 7/2 21 21 Right Triangle

What have you learned today?

What is still confusing?

5-3 Concurrent Lines, Medians, and Altitudes

Objective:• To identify properties of perpendicular

bisectors and angle bisectors• To Identify properties of medians and altitudes

Key ConceptPerpendicular Bisectors Altitudes circumscribe

MediansAngle Bisectors inscribe

Key Concepts

Medians –AD = AG + GDAG = 2GD

E F

D

Try

• Give the coordinates of the point of concurrency of the incenter and circumcenter.

• Angle bisectors ( 2.5,-1)• Perpendicular bisectors• (4,0)

Try

• Give the coordinates of the center of the circle.

• (0,0) perpendicular bisectors.

Determine if AB is an altitude, angle bisector, median, perpendicular bisector or none of these?

• perpendicular bisector median

none

angle bisector altitude

What have you learned today?

What is still confusing?

7-5 Proportions in Triangles

Obj: To use the Side-Splitter Theorem and Triangle-Angle Bisector Theorem.

Side-Splitter Theorem

If a line is parallel to a side of a triangle and intersect the other two sides, then this line divides those two sides proportionally.

Side-Splitter Theorem

Triangle-Bisector Theorem

if a ray or segment bisects an angle of a triangle then divides the segments proportionally.

Triangle-Angle Bisector Theorem

Example 1Find the value of x. 24 40

x 30

24 = x 40 30

720 = 40x

18 = x

Example 2

Find x and y.

6 5

x 12.5

9 y 6 = 5 x = 12.5X 12.5 9 yX = 15 y = 7.5

What have you learned today?

What is still confusing?

5-5 Inequalities in Triangles

Objective:• To use inequalities involving angles of triangles• To use inequalities involving sides of triangles

Key Concepts

• Triangle inequality – the sum of two sides is greater than the third side.

Try

• Order angles from least to greatest.B, T, A

• Order the sides from lest to greatest. BO, BL, LO

Try

Can the triangles have the given lengths? Explain.

yes 7 + 4 > 8 yes 1 + 9 > 9 yes 1.2 + 2.6 < 4.9 no

Try

Describe possible lengths of a triangles.

4in. and 7 in

7 – 4 7 + 43 < third side length < 113 < x < 11

What have you learned today?

What is still confusing?

Simplifying Radicals

• √ radical• Radicand – number inside the radical

• http://www.youtube.com/watch?v=HU5IawUD2o8

• You can click on other videos for more explainations.

√

Examples

1) √6 √8∙

√2 2 2 2 3∙ ∙ ∙ ∙ 4√3

2) √90 √2 3 3 5∙ ∙ ∙

3√10

3) √243 √3 √3 3 3 3 3∙ ∙ ∙ ∙ √39 √3 √3

9

Division – multiply numerator and denominator by the radical in the denominator

4) √25 √3

5 √3∙ √3 √3∙ 5 √3∙ 35) 8 = √14 √ 28 7

6) √5 √35∙ √14√5 5 7∙ ∙ √2 7∙5√7 √2 7∙√2 7 √2 7∙ ∙35 √2 = 5 √214 2

What have you learned today?

What is still confusing?

Chapter 8-1 Pythagorean Theorem and It’s Converse

Objective: to use the Pythagorean Theorem and it’s converse.

c2 = a2 + b2

Pythagorean Triplet

Whole numbers that satisfy c2 = a2 + b2.Example: 3, 4, 5 Can you find another set?

Ex 1 Find the value of x. Leave in simplest radical form.

Answer: 2 √11

x 12

10

Ex 2: Baseball

A baseball diamond is a square with 90 ft sides. Home plate and second base are at opposite vertices of the square. About far is home plate from second base?

About 127 ft

Pythagorean Theorem

Acute c2 < a2 + b2

Right c2 = a2 + b2

Obtuse c2 > a2 + b2

B

a c

C b A B

a c

C b A

B

a c

C b A

Ex 3:Classify the triangle as acute, right or obtuse.

a) 15, 20, 25rightb) 10, 15, 20Obtuse

What have you learned today?

What is still confusing?

Ch 8-2 Special Right Triangles

Objective:To use the properties of 45⁰ – 45⁰ – 90⁰ and 30⁰ – 60⁰ - 90⁰ triangles.

45⁰ – 45⁰ – 90⁰ 30⁰ – 60⁰ - 90⁰ x - x - x√2 x - x√3 - 2x

Special Right Triangles

45⁰ – 45⁰ – 90⁰ 30⁰ – 60⁰ - 90⁰

Example 1

Find the length of the hypotenuse of a 45⁰ – 45⁰ – 90⁰ triangle with legs of length 5√6 .

45⁰ – 45⁰ – 90⁰ x - x - x√2X = 5√6 x√2 = 5√6√2 substitute into the formula = 10 √3

Example 2

Find the length of a leg of a 45⁰ – 45⁰ – 90⁰ triangle with hypotenuse of length 22.

45⁰ – 45⁰ – 90⁰ x - x - x√2x√2 = 22 solve for xX = 22 = 22√2 = 11√2 √2 2

Example 3:

The distance from one corner to the opposite corner of a square field is 96ft. To the nearest foot, how long is each side of the field?

45⁰ – 45⁰ – 90⁰ x - x - x√2x√2 = 96 solve for xX = 96 = 96√2 = 48√2 √2 2

Example 4

The longer leg of a 30⁰ – 60⁰ - 90⁰ triangle has length of 18. Find the lengths of the shorter led and the hypotenuse.

30⁰ – 60⁰ - 90⁰ x - x√3 - 2xx√3 = 18 solve for xX = 18 = 18√3 = 6√3 – short leg √3 3 12√3 - hypotenuse

Example 5

Solve for missing parts of each triangle:x = 10

y = 5√3

5

y x

What have you learned today?

What is still confusing?

7-4 Similarities in Right Triangles

Objective: To find and use relationships in similar right triangles

• Geometric mean with similar right triangles

Example 1

Find the Geometric Mean of 3 and 15.√3 15 ∙3 √ 5

Find the geometric mean of 3 and 48.√3 48 ∙12

Example2 Find x, y, and z.

X = 69 x36 = 9x

4 = x

9 = z z 9+x

Z ²= 9(13)

Z = 3√13

y = x 9+x y

Y ² = 4(13)

Y = 2√13