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Math 310 Section 10.4 Similarity Similar Triangles Def ΔABC is similar to ΔDEF, written ΔABC ~ ΔDEF, iff <A is congruent to <D, <B is congruent to <E, <C is congruent to <F and AB/DE = AC/DF = BC/EF Ex The two following triangles are similar: ΔABC ~ ΔDEF. E 70° 8 4 B 4 A 2 70° 80° 30° 3 C D 80° 30° 6 F AA Property Thrm If two angles of one triangle are congruent, respectively, to two angles of a second triangle, then the triangles are similar. Denoted: AA Note: sometimes called the AAA property. Ex B 4 85° A 45° 6 Are the two triangles similar? If they are find the remaining sides. C E 8 85° 6 D 45° F Theorem 10-4 Thrm If a line parallel to one side of a triangle intersects the other sides, then it divides those sides into proportional segments. C E D A B Ex B m BE = 1.68 m ED m AB = 3.93 cm m BE = 3.59 cm A = 1.68 m AC E m AC = 2.33 cm m ED = 2.13 cm C m AB D Ex Suppose line DE is parallel to line segment BA in triangle ABC. If ratio of BD to DC is 2/3 and CE is length 3, what is the length of AE? C D E B 2 A Theorem 10-5 Thrm If a line divides two sides of a triangle into proportional segments, then the line is parallel to the third side. C E D A B Ex Find the measures of all the interior angles of triangle ABC. C 6 D 4 B 35°, 105 °, 40 ° 3 35° 105° 2 A E Theorem 10-6 Thrm If parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on any transversal. 3 3 2 2 Ex 5 7 ? 5 Given the three lines are parallel, what is the length of the segment next to the question mark? 7 Triangle Midsegment Def The midsegment of a triangle connects the midpoints of two adjacent sides of the triangle. Midsegment Theorem Thrm The midsegment is parallel to the third side of the trianlge and half as long. F I J G H Ex F 60° J I H 50° 8 G 50°, 60°, 70°, 4 Given that JI is the midsegment of triangle FGH, find all the interior angles of the triangle and the length of the midsegment. Theorem 10-8 If a line bisects one side of a triangle and is parallel to a second side, then it bisects the third side and therefore is a midsegment. F J 7 G 7 25° I 25° H Ex H I 110 ° F 30° 40° 3 G J 3 Is the segment JI the midsegment of triangle FGH? Indirect Measurement One practical use of these theorems is the ability to measure objects and distances that would be impossible or impractical to do directly. Ex Book: pg 689.