Download Plane Geometry - Oasis

Plane Geometry
ACT Review
It’s all a matter of degree
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Degrees in a circle Degrees in a line Degrees in a right angle –
Degrees in a triangle –
Degrees in a quadrilateral -
360
180
90
180
360
To Scale or Not to Scale?
• That is the question. In the instructions for
the ACT, the test writers say that the
diagrams are “NOT” necessarily drawn to
scale.
• Not one test has ever had a diagram not
drawn to scale. So use this fact to help
you on the test.
Example 1
• How big is the angle?
• Obviously, you don’t know exactly how big this angle is,
but it’s easy to compare it with an angle you do know.
Estimate.
Example 2
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In the figure, O,N, and M are collinear. If the lengths of ON and NL are the
same, and the measure of angle LON is 30 and the measure of LMN is 40,
What is the measure of angle NLM?
O
30°
N
40°
L
M
A.
40
B
80
C
90
D
120
E
150
Let’s Do It Again
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In the figure, if AB = 27, CD = 20, and the area of triangle ADC = 240, what
is the area of the polygon ABCD?
F. 420
G. 480
H. 540
J. 564
B
K 1,128
C
27
20
A
D
Angles and Lines Review
A
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Line
Collinear
Line Segment
Supplementary
Complementary
Perpendicular
Parallel
B
C
Angles Review
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Name the supplementary and vertical angles
B
A
100
C
y
x
D
What do all 4 angles add up to?
100
Angles Review (cont’d.)
If angle A = 110, what’s the measure of the others?
A B
D C
E
F
H
G
TRIANGLES
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Triangle Sum Theorem
Isosceles
Right
Equilateral
Pythagorean Theorem
Size of angles compared to length of sides
Triangles (cont’d.)
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There are 3 very common Pythagorean triples used on the ACT’s
3 - 4 - 5 (and it’s multiples)
5 - 12 - 13
7 - 24 - 25
Don’t Waste Time!
• Is this a 3-4-5 triangle?
4
3
• Is this a 5-12-13 triangle?
12
5
2 Other Right Triangles the ACT uses
• Right Isosceles:
5√2
5
3√2
3
3
5
• The 30-60-90 Triangle
10
5√3
8
4
5
4√3
Triangles (cont’d.)
• In the isosceles triangle below, are the sides = 3√2 ?
3
x
x
• In the right triangle, does x = 4√3 ?
x
4
4
Area of a Triangle
A = ½ bh
Height is measured as the perpendicular distance from
the base of the triangle to its highest point.
Similar Triangles
• Corresponding angles are congruent
• Sides are proportional.
60
x
4
4
2
30
2√3
4√3
ACT Triangle Problems
• Most of the triangle problems on the ACT
combine several of the triangle concepts we
just reviewed. Be flexible, and look for
clues as to which concepts are being
tested.
ACT Triangle Problems (cont’d.)
• In the figure, O,N, and M are colinear. If the length of
ON and NL are the same, and the measure of <LON is
30 degrees, and <LMN is 40 degrees, what is the
measure of < NLM?
O
30°
N
40°
M
ACT Triangle Problems (cont’d.)
• Square ABCD is attached to triangleADE. If <EAD is 30
degrees, and segment AE = 4√3, then what is the area
of square ABCD?
B
A
30
C
4√3
E
D
4 sided figures
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Area, perimeter, sum of angles
Rectangle
Square
Parallelogram
Trapezoid
Circles
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Radius
Diameter
Chord
Area
Circumference
Tangent line
• If area of a circle is 16 sq. meters, what’s the
radius in meters?
Circles (cont’d.)
• In the figure, the circle with center O is inscribed
inside a square. If a side of the square
measures 8 units, what is the area of the shaded
region?
F. 8-16л
G. 8 л
H. 16 л
J. 64 л
K. 64 - 16 л