Unit triangles
Sponsored Links
This presentation is the property of its rightful owner.
1 / 87

Unit: Triangles PowerPoint PPT Presentation


  • 91 Views
  • Uploaded on
  • Presentation posted in: General

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:

Download Presentation

Unit: Triangles

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


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

C + B = BAD

1 2

Remote interior angles

Exterior angle


Example 1Using the Exterior Angle Theorem

63+56 = 119

X = 119


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

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


Polygon Names


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


Key Concept

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


Example 1

  • 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


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?

  • 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?


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

  • √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?


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

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


  • Login