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Unit: Triangles

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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

C + B = BAD

1 2

Remote interior angles

Exterior angle

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.

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 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?

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 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

- 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 = 426X = 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 have you learned today?

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 –

AD = AG + GD

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.

- 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.

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

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

Triangle-Angle Bisector Theorem

Find the value of x.

24 40

x 30

- 24 = x
- 40 30
- 720 = 40x
- 18 = x

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?

Objective:

- To use inequalities involving angles of triangles
- To use inequalities involving sides of triangles

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

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

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

Can the triangles have the given lengths? Explain.

yes 7 + 4 > 8 yes

1 + 9 > 9 yes

1.2 + 2.6 < 4.9 no

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?

- √ radical
- Radicand – number inside the radical
- http://www.youtube.com/watch?v=HU5IawUD2o8
- 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

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?

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

c2 = a2 + b2

Whole numbers that satisfy c2 = a2 + b2.

Example: 3, 4, 5

Can you find another set?

Answer: 2 √11

x 12

10

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

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

- 15, 20, 25
right

b) 10, 15, 20

Obtuse

What have you learned today?

What is still confusing?

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

45⁰ – 45⁰ – 90⁰ 30⁰ – 60⁰ - 90⁰

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

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

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

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

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

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