GeoBlog / 2006-07 / 2nd Period

 Formulas:

a²= b²+c²-2bc(cos<A)

b²= a²+c²-2ac(cos <B)

c²= a²+ b²-2ab(cos<C)

Use the law of Cosines when:

-you’re given three sides

-you’re given two sides and and <

Ex 1:

x=? y=11 z=25 <x= 45

x²=25²+11²-2(25)(11) cos 45

x²= 625-2(275) cos 45

x²= 625-550 (cos 45)

x= 18.9

Ex 2:

<L=?

24²=5²+27²-2(5)(27) cosL

576=25+729-770 (cosL)

178=270(cosL)

<L= 48.8

If parallelogram:
**(both) If opposite sides are parallel, then p’gram
**(both) If opposite sides are congruent, then p’gram
**(both) If opposite angles are congruent, then p’gram
**If diagonals bisect, then p’gram
** If one pair of opposite sides are parallel and congruent, then p’gram

Example 1:

Determine whether this quadrilateral is a parallelogram. Justify your answer.
Yes, if opposite angles are congruent, then p’gram.

Example 2:
 Yes, if diagonals bisect, then p’gram.

-Tonights homework is Pg. 421 #13-24 and Pg. 769;8-3 #1-6
 

Parallelogram: A quadrilateral where opposite sides are parallel

Characteristics

1. Opposite sides are parallel
2. Opposite sides are congruent
3. Opposite angles are congruent
4. Consecutive/ Adjacent angles are supplementary
5. Diagonals bisect each other

<>** In a parallelogram, the diagonal divides the parallelogram into two congruent triangles

The triangles are congruent by ASA or SSS

<>

<>Example one: RSTU is a parallelogram. Find angle URT, RST and y.
angle URT equals 40 degrees because of alternate interior angles
angle RST equals 122 degrees because consecutive angles are supplementary
y equals 6 because 3y=18, set them equal because opposite sides are congruent

<>Example two: Find angle BDC , angle BCD, and x
x= 5 because you set 4x equal to 20 because opposite sides are congruent
angle BDC equals 54 degrees because alternate interior angles are congruent
angle BCD equals 62 degrees because consecutive/ adjacent angles are supplementary

The homework is page 415, problems 16-31 and 35-36

An angle of elevation is the angle between the line of sight and the horizantal when a observer looks upward.

An angle of depression is the angle between the line of sight when an observer looks downward, and the horizontal.

The law of sines is the application of trig functions with non-right triangles.

Trig Functions

Sine=sin=  opp

                  Hyp            Sin of angle a = a                      Sin of angle c= y

                                                              z                                             a                       

Cosine=cos=Adj                           Cos of angle a=x                 Cos angle c=y                                                                                  

                    Hyp                                                   z                                     z

Tangent of tan=Opp      Tan of angle a= y                      Tan of angle c= x

                        Adj                                x                                              y

When to use the 2^nd key

If you have angle measure? Looking for a missing side?

            Use sin/tan of the angle

If you have the side measures? Looking for angle measures?

            Use inverse sin/cos/tan

Ex 1.

Sine of 7= y      y=(sin 7)60

                 60   y= 7.31

Ex.2

S=cos of 41= x

                      32

X= .6157

Ex.3

Sin x=9

          29          x= 18.080

** If 2 triangles are similar, then the perimeters are in proportion to the corresponding sides. 

**If 2 triangles are similar then altitudes are proportional to the corresponding sides.

** If triangles are similar then angle bisectors are proportional to the corresponding sides.

** If triangles are similar then the medians are proportional to the corresponding sides.

** an angle bisector divides the opposite sides of a triangle into segments proportional to the other 2 sides.

Ex. 1

Ex. 2

11/30/06  Proportion 

Ratio- comparison of 2 quantities

           A:B     or     A/B

Proportion- 2 equivalent ratios
Example:  A/B= C/D
In the proportion A/B = C/D,
Means -> B and C
Extreme -> A and D

12/1/06 Similarity

Similar(~) – Angle are congruent sides in proportion

Scale factor: relationship between 2 ~ figures

SAS

** If 2 sides of a triangle are congruent and incleded angle is greater, then the opposite side is larger.

If angle 1 > angle 2, then a>b.

 SSS

** If 2 sides of a triangle are congruent and the third side is longer, then the opposite angle

 If a>b, then angle 1 > angle 2.

 Ex:

x=?

  2.5>x>6 because x+5>3x-7, x=5>0, and 3x-7>o