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

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

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

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

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

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

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

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

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) 2 2

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 2 2 2 2

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)

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

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