final exam key concepts. vertical angles find the value of “x”. 68 = 2x + 32 36 = 2x 18 = x

Final Exam

Key Concepts

Vertical Angles

• Find the value of “x”.

(2x + 32)68

68 = 2x + 32

36 = 2x

18 = x

Segments and Lengths• In the diagram, and S is the

midpoint of . QR = 4, and ST = 5. Find the following values.

1. RS =

2. PR =

3. PQ =

PR RTRT

5

P Q R S T10

6

Parallel Lines

• Find the missing values.

z

y

x

110

70o

70o

110o

Straight Line is 180o

• Find the difference between the larger angle and the smaller angle.

(4x + 19)

(3x)3x + (4x + 19) = 180

7x + 19 = 180

7x =161

x =23

3(23) = 69 and 4(23) + 19 = 111

111 – 69 = 42

Angles of Triangle = 180o

• The angles of a triangle are in the ratio of 1:2:3. Find the measure of each angle.

1x + 2x + 3x = 180

6x = 180

x = 30

30o, 60o, 90o

Exterior Angle Theorem

• Find the missing value.

x

Note: Figure not drawn to scale.

35

25

35o + 25o = 60o

60o

Angles and Sides of a Triangle

• Remember, largest angle is OPPOSITE of longest side….

• And smallest angle is OPPOSITE of smallest side.

45

67

68

B

C

A

BC

AB

TRIANGLE INEQUALITY THEOREM

• The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

• Is it possible for a triangle to have the following lengths?

3, 6, 8 6 + 3 = 9 > 8 YES

Three Theorems about Triangles

1) If c2 = a2 + b2, then the triangle is a right triangle.

2) If c2 < a2 + b2, then the triangle is an acute triangle.

3) If c2 > a2 + b2, then the triangle is an obtuse triangle.

12

RIGHT TRIANGLES

Special Right Triangles

Pythagorean Theorem

62 + 72 = x2

36 + 49 = x2

85 = x2

85x

Pythagorean Triples

3, 4, 5

5, 12, 13

8, 15, 17

7, 24, 25

Quadrilaterals

Quadrilaterals

• Complete the worksheet.

5 Ways to Prove Parallelograms

1. Both pairs of opposite sides are parallel

2. Both pairs of opposite sides are congruent

3. Both pairs of opposite angles are congruent

4. One pair of opposite sides congruent and parallel

5. Diagonals bisect each other.

INTERIOR MEASURES

SUM of the INTERIOR

Measures of Any Polygon

(n – 2)180o

(4-2)180o = 360o (8-2)180o = 1080o

EXTERIOR ANGLES

Sum of the measures of the EXTERIOR angles of any polygon

= 360o

e s

r

o

h

p

a

r

c

h + o + r + s + e = 360oc + r + a + p = 360o

REGULAR POLYGON

A polygon that is both equilateral and equiangular.

Areas

2

1 2

1 2

AREA

Rectangles =

Squares =

Parallelogram

1Triangles =

21

Rhombus= 2

1Regular Polygon =

21

Trapezoids =2

bh

s

bh

bh

d d

ap

h b b

Circles

2

2

CIRCLES

Area of Circle =

Circumference of Circle = 2

Arc length = 2360

Area of a sector = 360

r

r

xr

xr

Angles in a Circle

central angles =

1inscribed angles =

21

interior angles = ( )21

exterior angles = ( )2

arc

arc

big small

big small

80o

50o

50o

30o

Lengths in a Circle

12(9) = 18x

108 = 18x

6 = x

Lengths in a Circle

3(8) = 2(12)

24 = 24

Lengths in a Circle

122 = x(x+12+x)

144 = 2x2 + 12x

x = 8

SAT Formulas