Quarter Three Math Reviewer

Math Q3 (early) *use at your own risk! *sponsored by celine's notebook
Legend:
/: divided or OVER
*:multiplied
+:added
>:greater than
<:lesser than
( : ) : is to
Get into the habit of replacing these symbols with words in your head.

#Inequality
-Transitive Property:
1. If a>b and b>c, then a>c.
2.If a<b and b<c, then a<c.

-Addition and Subtraction Properties
1. If a>b, then (a+c)>(b+c) and (a-c)>(b-c)
2. If a<b, then (a+c)<(b+c) and (a-c)<(b-c)

-Multiplication and Division Properties
1. If c>0 and a<b, then (a*c)<(b*c) and (a/c)<(b/c)
2. If c>0 and a>b, then (a*c)>(b*c) and (a/c)>(b/c)
3. If c<0 and a<b, then (a*c)>(b*c) and (a/c)>(b/c)
4. If c<0 and a>b, then (a*c)<(b*c) and (a/b)<(b/c)
The properties of inequality are confusing. Take them slowly and put each math sentence into words to help you out.
#Theorems of Triangles
-Theorem 5 - 9
[If one side of a triangle is longer than another side, then the angle opposite the longer side is greater than the angle opposite the shorter side.]

Measure of side a: 90
Measure of side b: 25
Measure of side c: 70


-Theorem 5 - 10
[If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle.]

Measure of side a: 90
Measure of side b: 25
Measure of side c: 70




-Triangle Inequality Theorem
[The sum of the lengths of any two sides of a triangle is greater than the length of the third side.]

Measure of side a: 90
Measure of side b: 25
Measure of side c: 70
Is this a triangle? Let us check.
90+25? 115. 115 is more than 70.
25+70? 95. 95 is more than 90.
70+90? 160. 160 is more than 25.
Yes, it is a triangle!

Now for a word problem:
If the lengths of the two sides of a triangle are 10 and 15 between, what two numbers must the measure of the third side fall?
The answer is between 5 to 25.
How do you get it?
10+15 is 25.
Because: 10+15 MUST BE GREATER THAN the missing side.
15-10 is 5.
Because: 15-10 MUST BE LESSER THAN the missing side.
Working equation (where x is missing side) : 10+15>X>15-10>10-15
25>X>5>-5 Final Answer: 25>x>5


#Quadrilaterals

A Parallelogram is either a Rhombus, Square, or Rectangle.
A Trapezoid is never a parallelogram.
A Rhombus is sometimes a square but always a Parallelogram.
A Square has both the properties of a Rhombus and Rectangle.
A Rectangle is sometimes a square but always a Parallelogram.






Parallelogram: a quadrilateral with both pairs of opposite sides parallel.
Rectangle: a quadrilateral with 4 right angles.
Rhombus: a quadrilateral with four congruent sides.
Square: a quadrilateral with four congruent sides and four right angles.
Trapezoid: a quadrilateral with exactly one pair of parallel sides.
Isosceles Trapezoid: A trapezoid with congruent legs
#Quadrilateral Theorems
~Parallelograms (Shared theorems)
-Opposite angles in a parallelogram are congruent.
-consecutive angles in a parallelogram are supplementary.
-opposite sides in a parallelogram are congruent
-Diagonals of parallelograms bisect each other
-If the diagonals of a parallelogram are perpendicular then the parallelogram is a rhombus
I call these shared theorems because rhombuses, squares and rectangles are all parallelograms. Since they are all parallelograms, these theorems always apply to them. ( remember: trapezoids are not parallelograms)

~Rectangles
-Diagonals of rectangles are congruent

~Rhombuses
-Diagonals of rhombuses are perpendicular
-If the diagonals of a parallelogram are perpendicular then the parallelogram is a rhombus
-Each diagonal of a rhombus bisects a pair of opposite angles

~Trapezoids
The median is the segment that joins the MIDPOINTS of the LEGS.
-The median of a trapezoid is parallel to the bases and its measure is one half the sum of the two bases.

~Isosceles Trapezoid
-The diagonals of an isosceles trapezoid are congruent.
-Both pairs of base angles of an isosceles trapezoid are congruent.
# Ratios and Proportions
Definitions:
Ratio- comparison of two numbers using division (expressed as a/b or a:b) (only if b is not 0)
Proportion: an equation of the form ( a/b is equal to c/d) that states two ratios are equivalent
Means/Extremes: in a/b and c/d, b and c are the means. a and d are the extremes. (think of it as an X)
Cross Products: in a/b and c/d, b*c and a*d are the cross products. The proportion is true and only true if the cross products are equal. ( only if b is not =0 and d is not = 0)

Similar Polygons: two polygons are similar if and only if their corresponding angles are congruent and the measure of their corresponding sides are proportional.
Scale Factor: ratio of the lengths of two corresponding sides of two similar polygons

Similarity Theorems (triangles):
Side Side Side SIMILARITY THEOREM (SSS)
-If the corresponding side of 2 triangles are proportional, then the triangles are similar.
Side Angle Side SIMILARITY THEOREM (SAS)
-If two sides of a triangle are proportional to the two sides of another triangle, and if both sides' included angle is congruent, the triangles are similar.
Angle Angle SIMILARITY THEOREM (AA)
-If two angles of a triangle are congruent to two corresponding angles of another triangle, the triangles are congruent.






When asked to:

Solve for X in proportional polygons?
Use cross products. Compare one pair of corresponding sides to another pair of corresponding sides. Make sure one pair has the missing variable. Also use this when asked to prove if polygons are similar.

Get the measure of each side/angle of a triangle?
You will always be given a ratio, like 8:4:3. Then you will be given a measure, such as "total measure of all sides is 253." just turn it into an equation like 8x+4x+3x=253. Work from there. For the measure of all angles, just remember the angle sum theorem: all angles in a triangle will add up to 180.