Plane Geometry

ACT Review

It’s all a matter of degree

• Degrees in a circle - 360• Degrees in a line - 180• Degrees in a right angle – 90• Degrees in a triangle – 180• Degrees in a quadrilateral - 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

• 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

N

ML

40°

30°

A. 40

B 80

C 90

D 120

E 150

Let’s Do It Again

• 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• K 1,128

20

27

A D

B

C

Angles and Lines Review

• Line• Collinear• Line Segment• Supplementary• Complementary• Perpendicular• Parallel

A B C

Angles Review• Name the supplementary and vertical angles

A

B

C

D

100

100x y

What do all 4 angles add up to?

Angles Review (cont’d.)

A

If angle A = 110, what’s the measure of the others?

B

CD

G

FE

H

TRIANGLES

• Triangle Sum Theorem• Isosceles• Right• Equilateral• Pythagorean Theorem• Size of angles compared to length of sides

Triangles (cont’d.)

• 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?

• Is this a 5-12-13 triangle?

34

5

12

2 Other Right Triangles the ACT uses

• Right Isosceles:

• The 30-60-90 Triangle

3

3

3√2

4

5

5

5√2

4√3

8

5

105√3

Triangles (cont’d.)

• In the isosceles triangle below, are the sides =

• In the right triangle, does x =

3√2 ?

4

4√3 ?

x

x

3

x4

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.

244

x

2√3 4√3

30

60

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

N

M40°

30°

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?

4√3

A

B

C

DE

30

4 sided figures

• Area, perimeter, sum of angles• Rectangle• Square• Parallelogram• Trapezoid

Circles• 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