Week 3: GEOMETRY – Part I

SAT Prep

I. LINES and ANGLES

A.) Terminology and Notation

Lines / Rays / Segments 

Angles – Classification Straight - 180°Vertical - = Circle – 360°

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Ex. In the figure, what is the value of a? 

3 2 a a b

3 180 a b

Ex. In the figure below R, S, and T are all on line l. What is the average of a, b, c, d, and e?

b c

18036

5

ab c

de l

R S T

3a

cb

(a+2b)

180 3 b a

3 2 180 3 a a a

3 360 5 a a

8 360a

45a

Ex. In the figure, what is the value of x?  

   3 10 5 10 x x

2 20x

3 10 5 2 x x(3x+10) 5(x - 2)

10x

Ex. Line m bisects AOB, what is the value of x?

5025

2 x

180 130 50 AOCl

m

k

O x

130

A

B

B.) Parallel Lines – 4 pair of congruent angles. – Know angle theorems. 

Perpendicular to Parallels Thm.

Ex. In the figure below l // m, find the value of x.   

by Alternate Interior Angles Thm.

40 x

40

x

140 l

mk

Ex. Given AB // CD in the figure below, find the value of x.

Ex. Given the figure below l // m, find the value of a + b.

x

37

BA

DC

37 by Alternate Interior Angles Thm.

90 37 53x

45

a

bl

m

b

a 45a b

IN GENERAL – Sum of angles of all polygons = (n-2)180Sum of the exterior angles = 360 degrees

A.) Classification  

by angles : Acute Right Obtuse

by sides: Scalene Isosceles Equilateral

II. TRIANGLES

Ex. Given the figure below, find x.

 

Ex. Given the figure below, find a.   

25

90 25 65x

75 45 120a

12035

x

a

75

45

B.) Theorems 

1.) Exterior Angle Thm. 

2.) Largest Side is opposite Largest Angle. 

3.) Smallest side GREATER THAN the DIFF. of other two.

 4.) Largest side LESS THAN the SUM of the other two.

 5.) PYTHAGOREAN THEOREM –

 Know Triples Esp. 3 – 4 – 5 and multiples

2 2 2 a b c

C.) Special Right Triangles –

45º – 45º – 90º

30º – 60º – 90º

45

s

s

45

s2

x

x

2x 60

303

Ex. What is the area of a square whose diagonal is 10? 

 

 

 Ex. In the diagram, if BC = , what is the value of CD?  

2

5 2 50A

6

10 105 2

2

630

45

C

D

B

22 2

2 2 4

Ex. If the lengths of two sides of a triangle are 6 and 7, what are the possible values of the third side?     

7 6 7 6 x

1 13 x

D.) AREA of a TRIANGLE  

1.) ½ bh

2.)

3.) Equilateral =

Ex. Find the area of an equilateral triangle whose side is 10.    

2 3

4

s

1sin

2ab C

2 3

4s

A

210 3

4A

25 3A

Ex. An equilateral triangle with an area of has what perimeter?   

48 s

3 4 3 12 3 P

4 3 s

2 3

4s

A

2 312 3

4s

248 s

12 3

Ex. A triangular traffic island with a flat surface is formed by the intersection of three streets. Two of the sides of the islands have lengths of 6.4 meters and 10.8 meters. If the measure of the angle between these two sides is 55º, what is the area, in square meters, of the triangular surface of the island?

6.4sin 55 5.24 h6.4

h

1sin

2A ab C

OR

55

110.8 5.243 28.3

2 A

10.8

sin 556.4

h

110.8 6.4 sin 55 28.3

2 A

E.) SIMILAR TRIANGLES 3 pairs of = angles3 pair of proportional sides

  Ex. Given the figure below, find BC.

AB BC

DE DC

4

3 4x

3 16x

15

3x

3

4

4

A B

C

ED

III. QUADRILATERALS

Sum of angles = 360 degrees A.) Special Quads.

1.) Parallelograms – Opp. Sides = Opp. Sides //Opp. Angles = Cons. Angles supplementary2 diagonals bisect each other

2.) Rectangles – All properties of // -ogramsDiagonals = All 4 angles = 90 degrees

 3.) Squares –

All properties of rectanglesAll four sides =

B.) Area formulas – Parallelogram = bhRectangle = lwSquare = s2 or ½ d2

Ex. What is the length of each side of a square whose diagonal is 10?    10

5 22

s s10

s

Ex. The length of a rectangle is twice the width. If the perimeter of the rectangle is the same as the perimeter of a square with side 6, what is the square of the length of a diagonal of the rectangle?  

2l w

2 2P w l 2 2 24 8d

l

w

24 2 2 2w w

24 6w4w

8l

d

2 80d

80 4 5d

Ex. If AB = BC and DB = 5, then the area of ABCD = .

ABCD is a square

21

2A d

A B

CD

5

215 12.5

2A

The End!!!