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# Unit: Triangles - PowerPoint PPT Presentation

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

Objective:

To classify triangle and find the measure of their angles

To use exterior angle Theorem

Classify by angles:

Classify by sides:

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

Example 1Using the Exterior Angle Theorem

63+56 = 119

X = 119

90-55 = 35

86-55 = 31

180-(86+35) = 59

Find the measure of each angle.

62-25 = 37

180-(56+62) =62

180 – 62 = 118

OR

56 + 62 = 118

Classify the triangles.

• By its sides 18cm, 20 cm, 18cm

isosceles

b) By its angles 91,20 ,69

obtuse

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

Equilateral and equiangular

• 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

• 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 Angle Sum Theorem

180(n-2)

Polygon Exterior Angle Theorem:

Sum of all exterior angles is 360 degrees.

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

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

• 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

• 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.

Objective:

To use and apply properties of isosceles and equilateral triangles

• Isosceles Triangle Theorem

• Converse of the Isosceles Triangle

• Theorem

• If a segment, ray or line bisects the vertex angle, then it is the perpendicular bisector of the base.

• If a triangle is equilateral,

then it is equiangular.

• If the triangle is equiangular,

then it is equilateral.

• What is still confusing?

Objective:

To use properties of midsegments to solve problems

Midsegments –

DE = ½AB and

DE || AB

Find the perimeter of ∆ABC.

16+12+14 = 42

• 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 is still confusing?

Objective: To write ratios and solve proportions.

• RATIO- COMPARISON OF TWO QUANTITIES.

• PROPORTION- TWO RATIOS ARE EQUAL.

• EXTENDED PROPORTION – THREE OR MORE EQUILVANT 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

• 5 20

x 3

b) 18 6

n + 6 n

• 15 = 20x

• ¾ = x

• 18n = 6n +36

• 12n = 36

• n = 3

• 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.

Objective: to identify and apply similar polygons

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

• Similarity ratio – ratio of lengths of corresponding sides

• 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

Find PT and PR 4 = X

11 X+12

11X = 4X + 48

7X = 48

X = 6

PT = 6 PR = 18

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 =

• 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.

objective:

To use properties of perpendicular bisectors and angle bisectors

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

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

WY is the  bisector of XZ

4

7.5

9

Isosceles triangle

6y = 8y -7 7 = 2y y = 7/2

21

21

Right Triangle

What is still confusing?

Objective:

• To identify properties of perpendicular bisectors and angle bisectors

• To Identify properties of medians and altitudes

Perpendicular Bisectors Altitudes

circumscribe

Medians

Angle Bisectors

inscribe

Medians –

AG = 2GD

E F

D

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

• Angle bisectors ( 2.5,-1)

• Perpendicular bisectors

• (4,0)

• 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? perpendicular bisector or none of these?

What is still confusing?

### 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 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 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 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 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 perpendicular bisector or none of these?

Objective:

• To use inequalities involving angles of triangles

• To use inequalities involving sides of triangles

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 perpendicular bisector or none of these?

• You can click on other videos for more explainations.

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

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 radical in the denominator

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

c2 = a2 + b2

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. radical in the denominator

x 12

10

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?

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. 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 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 radical in the denominator

45⁰ – 45⁰ – 90⁰ 30⁰ – 60⁰ - 90⁰

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 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 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 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 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 radical in the denominator

Objective: To find and use relationships in similar right triangles

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

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