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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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3 4 parallel lines and triangle sum theorem
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
Classifying Triangle

Classify by angles:

Classify by sides:


Theorems
Theorems

Triangle-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

Remote interior angles

Exterior angle


Example 1 using the exterior angle theorem
Example 1Using the Exterior Angle Theorem

63+56 = 119

X = 119


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

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
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
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
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
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
Polygons

Polygon Angle Sum Theorem

180(n-2)

Polygon Exterior Angle Theorem:

Sum of all exterior angles is 360 degrees.


Example 1
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
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
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


Closure1
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
4-5 Isosceles and Equilateral Triangles

Objective:

To use and apply properties of isosceles and equilateral triangles


Isosceles triangle key concepts
Isosceles Triangle Key Concepts

  • Isosceles Triangle Theorem

  • Converse of the Isosceles Triangle

  • Theorem


Isosceles triangle key concepts1
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
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 did you learn today?

  • What is still confusing?


5 1 midsegments
5-1 Midsegments

Objective:

To use properties of midsegments to solve problems


Key concept
Key Concept

Midsegments –

DE = ½AB and

DE || AB


Try 1
Try 1

Find the perimeter of ∆ABC.

16+12+14 = 42


Try 2
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
7-1 Ratios and Proportions

Objective: To write ratios and solve proportions.


Vocbulary
VOCBULARY

  • RATIO- COMPARISON OF TWO QUANTITIES.

  • PROPORTION- TWO RATIOS ARE EQUAL.

  • EXTENDED PROPORTION – THREE OR MORE EQUILVANT RATIOS.


Properties of ratios
PROPERTIES OF RATIOS

a c is equivalent to: 1) ad = bc

b d 2) b d 3) a b

a c c d

4) a + b c + d

b c


Example 11
Example 1

  • 5 20

    x 3

    b) 18 6

    n + 6 n

  • 15 = 20x

  • ¾ = x

  • 18n = 6n +36

  • 12n = 36

  • n = 3


Example 21
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
7-2 Similar Polygons

Objective: to identify and apply similar polygons


Vocabulary1
Vocabulary

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

  • Similarity ratio – ratio of lengths of corresponding sides


Example 12
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 22
Example 2

Find PT and PR 4 = X

11 X+12

11X = 4X + 48

7X = 48

X = 6

PT = 6 PR = 18


Example 31
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
5-2 Bisectors in Triangles

objective:

To use properties of perpendicular bisectors and angle bisectors


Key concept1
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 11
Try 1

WY is the  bisector of XZ

4

7.5

9

Isosceles triangle


Try 21
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
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 concept2
Key Concept

Perpendicular Bisectors Altitudes

circumscribe

Medians

Angle Bisectors

inscribe


Key concepts
Key Concepts

Medians –

AD = AG + GD

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
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? perpendicular bisector or none of these?

What is still confusing?


7 5 proportions in triangles

7-5 Proportions in Triangles perpendicular bisector or none of these?

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


Side splitter theorem
Side-Splitter Theorem perpendicular bisector or none of these?

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
Triangle-Bisector Theorem perpendicular bisector or none of these?

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

Triangle-Angle Bisector Theorem


Example 13
Example 1 perpendicular bisector or none of these?

Find the value of x.

24 40

x 30

  • 24 = x

  • 40 30

  • 720 = 40x

  • 18 = x


Example 23
Example 2 perpendicular bisector or none of these?

Find x and y.

6 5

x 12.5

9 y

6 = 5 x = 12.5

X 12.5 9 y

X = 15 y = 7.5


What have you learned today? perpendicular bisector or none of these?

What is still confusing?


5 5 inequalities in triangles
5-5 Inequalities in Triangles perpendicular bisector or none of these?

Objective:

  • To use inequalities involving angles of triangles

  • To use inequalities involving sides of triangles


Key concepts1
Key Concepts perpendicular bisector or none of these?

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


Try perpendicular bisector or none of these?

  • Order angles from least to greatest.

    B, T, A

  • Order the sides from lest to greatest.

    BO, BL, LO


Try perpendicular bisector or none of these?

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 perpendicular bisector or none of these?

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? perpendicular bisector or none of these?

What is still confusing?


Simplifying radicals
Simplifying Radicals perpendicular bisector or none of these?

  • √ radical

  • Radicand – number inside the radical

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

  • You can click on other videos for more explainations.


Examples
Examples perpendicular bisector or none of these?

  • √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
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? radical in the denominator

What is still confusing?


Chapter 8 1 pythagorean theorem and it s converse
Chapter 8-1 Pythagorean Theorem and It’s Converse radical in the denominator

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

c2 = a2 + b2


Pythagorean triplet
Pythagorean Triplet radical in the denominator

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

Answer: 2 √11

x 12

10


Ex 2 baseball
Ex 2: Baseball radical in the denominator

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
Pythagorean Theorem radical in the denominator

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

  • 15, 20, 25

    right

    b) 10, 15, 20

    Obtuse


What have you learned today? radical in the denominator

What is still confusing?


Ch 8 2 special right triangles
Ch 8-2 Special Right Triangles radical in the denominator

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
Special Right Triangles radical in the denominator

45⁰ – 45⁰ – 90⁰ 30⁰ – 60⁰ - 90⁰


Example 14
Example 1 radical in the denominator

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 24
Example 2 radical in the denominator

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 32
Example 3: radical in the denominator

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
Example 4 radical in the denominator

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
Example 5 radical in the denominator

Solve for missing parts of each triangle:

x = 10

y = 5√3

x

y

5


What have you learned today? radical in the denominator

What is still confusing?


7 4 similarities in right triangles

7-4 Similarities in Right Triangles radical in the denominator

Objective: To find and use relationships in similar right triangles



Example 15
Example 1 radical in the denominator

Find the Geometric Mean of 3 and 15.

√3∙15

3 √ 5

Find the geometric mean of 3 and 48.

√3∙48

12


Example2
Example2 radical in the denominator

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


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