1. By: sylvia Wei and Vibha Shivakumar
UNIT 5: SIMILARITY
By: sylvia Wei and Vibha Shivakumar
THESE SHAPES ARE POLYGONS

2. RAtios
Similar figures will have an equal ratio which is the ratio of the corresponding sides
a ratio is the comparison of two numbers using division in its simplest form
examples: a, a:b, or a to b b

3. PROPORTIONS a Proportion is 2 ratios set equal to one another
e1: x+5 = x+3
a1: 5(x+5)=7(x+3)
5x+25=7x+21
4=2x
2=x

4. GEOMETRIC MEAN a1: √(3*27)= √81= 9 a2: 6=√12x = 36=12x = 3=x
If the means in a proportion are equal, either mean is called a geometric mean or mean proportional between the extremes
Geometric mean formula: √(a*r)
e1: find the geometric mean of 3 and 27
e2: 6 is the geometric mean between 12 and what other number?
a1: √(3*27)= √81= 9
a2: 6=√12x = 36=12x = 3=x

5. TRIANGLE PROPERTIES
AA~ two triangles with two congruent angles are similar
SAS~ if two triangles have two pairs congruent sides and one pair of congruent angles, then they are similar
SSS~ two triangles with corresponding and proportional sides are similar

6. TRIANGLE PROPERTIES (cont…)
third angle theorem- if two triangles have two congruent angle pairs, then the third angle pair is also congruent
triangle sum theorem- sum of a triangle's interior angles is 180°
e1: solve for x
a1:90+67+x=180
157+x=180
x=23
90°
67° x

7. TRIANGLE APPLICATIONS
interior angles- inside angles
exterior angles- outside angles
*INTERIOR ANGLES FOR TRIANGLES WILL ALWAYS ADD UP TO 180 (triangle sum theorem)*
*EXTERIOR ANGLES WILL ALWAYS ADD UP TO 360 FOR ANY SHAPE*
interior angle
exterior angle

8. REGULAR POLYGONS
Polygon: 3+ segments that intersect exactly 2 sides with one endpoint
Regular Polygon: All sides and angles are congruent
Convex: no sides contain points inside the slope
Concave: side can contain point inside the shape
Convex polygon
Concave polygon

9. POLYGON FORMULAS n= number of sides
sum of exterior angles for any polygon is always 360
sum of a polygon’s interior angles= (n-2)(180)
polygon’s diagonals= n(n-3) or (n-3)(n)

10. POLYGON FORMULA EXAMPLES
e1:find the interior angles of a pentagon
a1:(n-2)(180)= (5-2)(180)=540
e2:find the # of diagonals for a pentagon
a2:n(n-3)= 5(5-3)= 5(2)= 10= 5

11. CONNECTION TO OTHER UNITS
when you dilate a figure, the new figure is similar to the original figure (unit 9)
to find the slant height of a frustum, you must use a proportion(unit 11)

12. REAL WORLD APPLICATION
finding the distance from the sky to the ground
finding an object’s shadow
finding the length or height of objects

13. COMMON MISTAKES
you cannot use AA~ to prove congruence between two triangles
solution: don’t use AA~ to prove congruence
setting up the proportion wrong
solution: be more careful when reading the diagram
not cross multiplying when solving for a variable using proportions
SOlution: Remember that the = sign is between the proportion, not a * sign