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Transcript
Chapter 7: Proportions and Similarity 7.1- Proportions Make a Frayer foldable 7.1 Ratio and Proportion Ratio A comparison of two quantities using division 3 ways to write a ratio: a to b a b a:b Proportion An equation stating that two ratios are equal Example: a c b d Cross products: means and extremes Example: a c b d ad = bc a and d = extremes b and c = means Your Turn: solve these examples Ex: 3 21 x 6 Ex: x2 4 2 5 3 * 6 = x * 21 5(x – 2) = 2 * 4 18 = 21x 5x – 10 = 8 x = 18/21 5x = 18 x = 6/7 x = 18/5 x = 3 3/5 Your Turn: solve this example The ratios of the measures of three angles of a triangle are 5:7:8. Find the angle measures. 5x + 7x + 8x = 180 20x = 180 x=9 45, 63, 72 7.2 : Similar Polygons Similar polygons have: Congruent corresponding angles Proportional corresponding sides A Polygon ABCDE ~ Polygon LMNOP B L E M C D P N Ex: AB CD LM NO O Scale factor: the ratio of corresponding sides 7.3: Similar Triangles Similar triangles have congruent corresponding angles and proportional corresponding sides Z Y A C X B angle A angle X ABC ~ XYZ angle B angle Y angle C angle Z AB AC BC XY XZ YZ 7.3: Similar Triangles Triangles are similar if you show: Any 2 pairs of corresponding sides are proportional and the included angles are congruent (SAS Similarity) All 3 pairs of corresponding sides are proportional (SSS Similarity) Any 2 pairs of corresponding angles are congruent (AA Similarity) 7.4 : Parallel Lines and Proportional Parts If a line is parallel to one side of a triangle and intersects the other two sides of the triangle, then it separates those sides into proportional parts. A Y *If XY ll CB, then AY AX YC XB C X B 7.4 : Parallel Lines and Proportional Parts Triangle Midsegment Theorem A midsegment of a triangle is parallel to one side of a triangle, and its length is half of the side that it is parallel to *If E and B are the midpoints of AD and AC respectively, 1 then EB = 2 DC A E D B C 7.4 : Parallel Lines and Proportional Parts If 3 or more lines are parallel and intersect two transversals, then they cut the transversals into proportional parts A B C D E F AB DE BC EF AC BC DF EF AC DF BC EF 7.4 : Parallel Lines and Proportional Parts If 3 or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal If AB BC , then DE EF A B C D E F 7.5 : Parts of Similar Triangles If two triangles are X similar, then the perimeters are proportional to the measures of corresponding sides A B C Y perimeterABC AB BC AC perimeterXYZ XY YZ XZ Z 7.5 : Parts of Similar Triangles If 2 triangles are similar, If 2 triangles are similar, the measures of the corresponding altitudes are proportional to the corresponding sides the measures of the corresponding angle bisectors are proportional to the corresponding sides X A S M B C D Y W AD AC BA BC XW XZ YX YZ Z L R O N U MO MN LM LN SU ST RS RT T 7.5 : Parts of Similar Triangles An angle bisector in a If 2 triangles are similar, triangle cuts the opposite side into segments that are proportional to the other E sides then the measures of the corresponding medians are proportional to the corresponding sides. A BC AB CD AD G T B J H D C I F U GI GH GJ HJ TV UT TW UW V W G FG EF GH EH H