# Unit: Triangles - PowerPoint PPT Presentation

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Unit: Triangles. 3-4 Parallel lines and Triangle Sum Theorem. Objective: To classify triangle and find the measure of their angles To use exterior angle Theorem. Classifying Triangle. Classify by angles: Classify by sides:. Theorems. Triangle-Angle Sum Theorem:

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Unit: Triangles

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## Unit: Triangles

### 3-4 Parallel lines and Triangle Sum Theorem

Objective:

To classify triangle and find the measure of their angles

To use exterior angle Theorem

### Classifying Triangle

Classify by angles:

Classify by sides:

### Theorems

Triangle-Angle Sum Theorem:

Sum of the angles

is 180.

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

1 2

Remote interior angles

Exterior angle

63+56 = 119

X = 119

90-55 = 35

86-55 = 31

180-(86+35) = 59

### TRY

Find the measure of each angle.

62-25 = 37

180-(56+62) =62

180 – 62 = 118

OR

56 + 62 = 118

### Example 3

Classify the triangles.

• By its sides 18cm, 20 cm, 18cm

isosceles

b) By its angles 91,20 ,69

obtuse

### Try

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

Equilateral and equiangular

### Closure

• 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 polygons

To 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

convex

convex

concave

concave

convex

### Polygons

Polygon Angle Sum Theorem

180(n-2)

Polygon Exterior Angle Theorem:

Sum of all exterior angles is 360 degrees.

### Example 1

Pentagon

• Name the polygon by its sides

• Concave or convex.

• Name the polygon by its vertices.

• Find the measure of the missing angle

Convex

QRSTU

(5-2)180 = 540

130+54+97+130 = 411

540 – 411 = 129

### Example 2

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

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

360/7 = 51 3/7

### 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 angle

360/10 = 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

Midsegments –

DE = ½AB and

DE || AB

### Try 1

Find the perimeter of ∆ABC.

16+12+14 = 42

### Try 2:

• If mADE = 57, what is the mABC?

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 = bc

b d 2) b d3) a b

a c c d

4) a + b c + d

b c

• 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 fall

If 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 = 426X = 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 = 15

6 2.5

2.5X = 80

X =

### 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 Concept

Perpendicular Bisectors Altitudes

circumscribe

Medians

Angle Bisectors

inscribe

Medians –

AG = 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 1

Find 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 = 12.5

X 12.5 9 y

X = 15y = 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 + 4

• < third side length < 11

3 < x < 11

What have you learned today?

What is still confusing?

• You can click on other videos for more explainations.

• √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

√3

9 √3

√3

9

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

4) √25

√3

5 ∙√3

√3 ∙√3

5 ∙√3

3

• 8 = √14

√ 28 7

6) √5 ∙ √35

√14

√5∙5 ∙7

√2∙7

5√7 √2∙7

√2∙7 √2∙7

35 √2 = 5 √2

14 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?

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?

### Pythagorean Theorem

B

a c

C b A

Acute c2 < a2 + b2

Right c2 = a2 + b2

Obtuse c2 > a2 + b2

B

a c

C b A

B

a c

C b A

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

• 15, 20, 25

right

b) 10, 15, 20

Obtuse

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√2

X = 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√2

x√2 = 22 solve for x

X = 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√2

x√2 = 96 solve for x

X = 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 - 2x

x√3 = 18 solve for x

X = 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

x

y

5

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 = 6

9 x

36 = 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