Geometry Unit

1
Geometry Unit
2
Basic Geometric Vocabulary

• Points a spot in space .A
• Labeled Point A
• Line A never ending straight path.

A
B
g
- Labeled AB or BA
3

• Line Segment A line with two endpoints
• Labeled CD DC

C
D

• Ray A part of one line that extends from a
  point indefinitely in one direction.

E
F

• Labeled EF Not FE

4

• Plane A flat surface that extends in all
  directions
• Labeled plane G, plane H, or Plane I

I
G H

• Angle Two rays with a common endpoint
• Labeled lt ABC, ltCBA, OR ltB

B
A
C
5
Angles

• Exterior of angle
• Interior of angle
• On the angle
• Vertex of the angle the point in the angle
  where the two rays meet.

6

• Co linear two points on the same line
• Co-planer two points on the same plane

B
A
C
E
D
F
G
7
Types of Angles

• Acute less than 90 degrees
• Obtuse more than 90 degrees
• Right- exactly 90 degrees
• Straight exactly 180 degrees

8

• Adjacent angles two angles that sit next to
  each other. They share a side.
• Linear pairs two adjacent angles that form 180
  degrees.

9
Bisect cut into two EQUAL pieces.
If you bisect a 75 degree angle, what are the two
new angle measurements?
10
Complementary Angles Angles that when added
together form 90 degrees.
Supplementary Angles Angles that when added
together form 180 degrees.
B
A
C
D
ltABC AND ltCBD are Complementary
ltABD and ? Are Supplementary?
D
A
B
C
What is another name for Supplementary Angles?
11
Special Angles

• Vertical Angles Two pairs of opposite angles
  formed by intersecting lines - Congruent

C
A
B
E
D
ltABD is Vertical to ltCBE . Can you give me
another pair of Vertical Angles?
12
Adjacent Angles

• Angles that sit next to each other and share a
  common side.

13
Angle Addition Postulate
A
B
C
D
If ltABC is 30 degrees and ltABD is 95 degrees,
what is the measurement of ltCBD?
14
x
35 d
Find x.
15

• Congruent - Exactly the same. Congruent angles
  have the exact same measurement.
• Congruence can be shown with measurements or with
  slash marks.

45 m 45 m
16
Congruent Triangle Postulates

• Angle, Side, Angle ASA

45
45
45
45
Angle, Angle, Side AAS
35
35
32
32
17
Side, Side, Side - SSS
4 M
4 M
Side, Angle, Side - SAS
78 cm
78 cm
90
87 cm
87 cm
90
18
Are these congruent? Give the Rule
23 d
23 d
45 d 36d 45d d 36d
19
All 3 angles in a triangle always equal 180
degrees.
A
ltA is 30 degrees, lt B is 48 degrees, how many
degrees is ltC?
B
C
50
x
55
Find x.
90
65
20
Special Congruent Angles formed with parallel
lines and a transversal
1
2
4
3
5
6
7
8
Vertical Angles congruent angles lt1 and lt4
21
Adjacent Angles Angles that sit next to each
other and form 180 degrees. (Linear Pairs)

• lt1 and lt2 are adjacent angles. Find another pair
  of adjacent angles.

1
2
4
3
5
6
7
8
22
Cooresponding Angles Angles that sit in the
same position on opposite parallel lines. They
are Congruent.

• lt 1 and lt 5 are Cooresponding. They are on top
  of the parallel line and to the left of the
  transversal. They are Congruent!
• Find another pair of Cooresponding angles.

1
2
4
3
5
6
7
8
23
Alternate Interior Angles These are angles
inside the drawing that are across from each
other, over the transversal. They are Congruent!
lt 3 and lt6 are Alt. Int. Angles
1
2
4
3
5
6
7
8
Alternate Exterior Angles Angles Outside the
drawing that are across from each other, over the
transversal. They are Congruent. lt 1 and lt 8 are
Alt. Ext. Angles.
24
Consecutive Interior Angles angles on the
interior that sit next to each other form 180
degrees Angle 3 and angle 5
1
2
4
3
5
6
7
8
25
Triangle Names Each triangle has two names!

• Label by the sides
• All three congruent sides equilateral
• Two congruent sides Isosceles
• No congruent sides - Scalene

• Label by the angles
• One right angle Right Triangle
• One obtuse angle Obtuse Triangle
• All three acute angles- Acute Triangle

26
Name the triangle with TWO names
40 d
98 M
75 M
5 M
90 d
40 d
27
Quadrilaterals

• Square
• 4 Congruent Sides
• Opposite sides are parallel
• 4 Right Angles

• Rectangle
• Opposite sides are congruent and parallel
• All angles are 90 degrees

28

• Rhombus
• 4 congruent sides
• Opposite sides are parallel
• No right angles, 2 acute, 2 obtuse

• Parallelogram
• Opposite sides are congruent and parallel
• No right angles, 2 acute, 2 obtuse

29

• Trapezoid
• Only one set of parallel sides

30
Polygons
Regular Polygons Have all equal/Congruent Sides
Which triangle is a regular polygon?
4 Sided Figure Quadrilaterals 5 Sided Figure
Pentagon 6 Sided Figure Hexagon 7 Sided Figure
Heptagon 8 Sided Figure Octagon 9 Sided
Figure Nonagon 10 Sided Figure Decagon 12
Sided Figure - Dodecagon
Extra Credit Find out what an 11 sided figure
is called. Bring me proof from the internet or a
book.
31
Similar Shapes
Similar shapes have the same angles, but
different sides.
7.5 m
5m
3m
4m
6M
Why is there no postulate that says AAA for
congruence?
32
Similar Shapes

• Find ratios for corresponding sides

G
E
D
A
B
C
F
H
AB EF
33
Ratios must be equal to be similar

• 5 2
• 4
• Cross multiply
• 20 20
• Equation is true therefore, these shapes are
  similar!

5cm
2cm
10 cm
4 cm
34
Proportions to Find Missing Sides
X
7 M
3 M

• 3 X
• 6 7
• 21 6X
• 6
• 3.5 X

6 M
35
Circles

• Parts of a Circle
• Circumference The length around the circle
• Radius. A line segment that connects the center
  of the circle to one point on the circumference.
• Diameter A line segment that cuts the circle
  into two equal parts.
• Chord A line segment that cuts a circle into
  two pieces not necessarily equal.

36
Circle Formulas

• Circumference the length around a circle
• Area The space inside the circle

D Circumference
2
r
Area
3.14
Find the Circumference and the area.
5 cm
37
Circles Continued

• Find the circumference and area of a circle with
  a radius of 4.5 m.
• Find the diameter of a circle whose circumference
  is 80 M.
• Find a radius from the picture below.
• Find a chord from the picture below
• Find a diameter from the picture below.

D
B
C
A
38
Area of Polygons

• Rectangle/Square formula LW A

78
60 4/5

• Triangle formula
• ½ bh A

6
15
6.5
7
12
39
Area of Parallelogram formula bh area
15 m
10 m
56 m
Area of Trapezoid formula ½(b1 b 2)h
5 cm
10 cm
13 cm
12 cm
15 cm
40
Pythagorean Theorem
2
2
2
Formula a b c
This formula works with right triangles ONLY!
c
a
a and b are the legs of the triangle. c is the
hypotenuse.
b
41
Pythagorean Practice

• Find the missing sides. Round to the tenths
  place.

5 m
6
6 m
b
5
42
13 m
x
5 m
X - 6
30 cm
x

• Tell me if the following can be a right triangle?
• a 12, b 9, c 15
• a 8, b 10, c 13

43
Surface Area

• The area of all sides of the 3 dimensional
  figure.

3 m
3 m
3 m
All sides are rectangles therefore we use lw
area formula. Front and back 3 x 3 9 Top and
bottom 3 x 3 9 Two sides are 3 x 3 9 Total
area is 9 x 6 sides or 54 square meters
44
4 m
3m
5m
Area lw Top and bottom 5x3 15 Front and
back 5 x 4 20 Sides 3 x 4 12
Total Surface area 15 15 20 20 12 12
94 square meters
45
Area of ends are triangles so ½ bh area. Hint
the base and the height are always connected to
the right angle.
½ bh or ½ 3 x 4 6
Rectangle sides Lw area 3x7 21 4 x 7 28 5
x 7 35
5
Total area 6 6 21 28 35 96 square
units.
3
7
4
46
Find the area of the top and the bottom. They
are circles so use the formula r
area Top and bottom 3.14 x 2 12.56 Side is a
rectangle with circumference as the length and 12
as the width.
4 m
2
12 m
2
Circumference or pi x diameter
3.14 x 4 x 12 150.72
12 m
Surface area is 12.56 12.56 150.72 175.84
squared meters.
47
Volume of 3 Prisms a prism has rectangular
sides
L x w x h volume 5 x 5 x 5 125 cubic units
5
5
5
½ bh H Volume ½ (4 x 5) x 7 ½ (20) x 7 10 x 7
70 cubic units.
h 4
H 7
b5
48
Volume of Cylinders Cones
2
r h volume
12
7
d 7 r 3.5 3.14 x 3.5x3.5 x 12 461.68 cubic
units
2
1/3
r h volume
1/3 x 3.14 x 4 x 4 x 12 1/3 ( 602.88) 200.96
cubic units
25
12
4
49
Volume of Pyramids
1/3 b h volume
Base is a rectangle so 1/3 (bh)H 1/3 (2x4)x 6 1/3
(8)(6) 1/3 (48) 16 cubic cm
2 cm
6 cm
4 cm
50
volume 1/3 base x height Base is a triangle
so 1/3 (1/2 bh) x H
6 cm
1/3 ( ½ x 3 x 4) x 6 1/3 ( ½ x 12) x 6 1/3 (6) X
6 1/3( 36) 12 cubic cm
7 cm
4cm
3 cm
51
Trigonometric Measurements
Tangent Opposite
A
Adjacent
17
5
Tangent Ratios Tan ltA 12/5 Tan ltC 5/12
C
B
12
Find the degrees for ltA using Tangent Tan ltA
12/5 2.4 Use table on pg. 812 ltA is about 77
degrees
52
A
Find the measurement of ltA Tangent of ltA
58/55 1.0545 ltA is approximately 47 degrees
55 cm
C
B
58cm
53
Sine Opposite Cosine Adjacent
hypotenuse hypotenuse

• Find the Sine and Cosine ratios for ltR and ltS

R
R
61
13
5
11
S
T
60
S
T
12
ltR Sine 60/61 ltR Cosine 11/61 ltS Sine 11/61 ltS
Cosine 60/61
ltR Sine 12/13 ltR Cosine 5/13 ltS Sine 5/13 ltS
Cosine12/13
54
Using Trigonometric Ratios

• A boat is pulling a parasailer. The line to the
  parasailer is 800 ft long. The angle between the
  line and the water is about 25 degrees. Use
  Trigononmetric ratios to solve the following
  questions.
• How high is the parasailer.
• How far back is the parasailer.

55

• How high is the parasailer. - x
• How far back is the parasailer. - y

• To find X
• Opposite Sinelt25
• hypotenuse
• X .4226
• 800
• 800 X .4226 (800)
• 800
• X 338.08

• To find Y
• Adjacent Cos lt25
• Hypotenuse
• Y .9063
• 800
• 800 Y .9063 (800)
• 800
• Y 725.04

800 ft
x
y
25 degrees