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SMCHS Geometry B Mr. Ricks Chapter 7 Homework Assignments: Section 7.1 Section 7.2 Section 7.3 Section 7.4 Section 7.5 Section 7.6 p. 243 – 244: p. 246 – 247: p. 250 – 251: p. 256 – 257: p. 264 – 266: p. 271 – 272: Chapter 7 Review: Chapter 7 Extra Credit: Review) 1 – 13 all (CE), 1 – 20 all (WE) 1 – 11 all (CE), 1 – 20 all (WE) 1 – 9 all (CE), 1 – 21 all (WE) 1 – 11 all (CE), 1- 13 all (WE) 1 – 6 all (CE), 1 – 10 all (WE) 1 – 9 all (CE), 1 – 11 all (WE) p. 277 – 278: 1-24 all (Chapter review) p. 279 – 280: 1-14 all (Chapter test), 1 – 29 odd (Algebra Section 7-1: Ratio and Proportion The ratio of one number to another is the quotient when the first number is divided by the second. Ratios are used to compare two numbers. You must use the same units (cm, m, yard, etc.) for the ratio to be accurate. Examples: 4 to 1 4:1 4 1 These are all examples of a single ratio written different ways. 20 m and 65m. p.h are examples of ratio that measure rate of change, or speed. Since the s ratios use different units, miles vs. meters and seconds vs. hours, you cannot accurately compare the ratios. A proportion is an equation stating that two ratios are equal. Examples: 1 9 is an example of a proportion. In this case, you have true statement. 3 27 Section 7-2: Properties of Proportions We can cross multiply proportions. This allows us to solve for a missing number in a proportion using the other three. 4 x 3 12 3 x 412 10 4 n 9 4n 910 3 x 48 4n 90 x 16 n 22.5 Section 7-3: Similar Polygons Two polygons are similar if their vertices can be paired so that: a) Corresponding angles are congruent b) Corresponding sides are in proportion (their lengths have the same ratio) F G B C E A D m AB = 3.00 cm H m EF = 4.25 cm ABCD is a sqaure. EFGH is a square. ABCD is similar to EFGH (written as ABCD EFGH) because: 1) the corresponding angles are congruent. 2) the corresponding sides are in proportion. What is the scale factor for the above picture? _____ If two polygons are similar, then the ratio of the lengths of two corresponding sides is called the scale factor of the similarity. J M L N I K m JI = 5.00 cm IJK and IJK and IJK and because: m M L = 2.50 cm LMN are equilateral triangles. LMN are also __________ triangles. LMN are similar triangles (written as IJK LMN) 1) the corresponding angles are congruent. 2) the corresponding sides are in proportion. What is the scale factor for the above picture? _____ Section 7-4: A Postulate for Similar Triangles Postulate 15: AA Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. O P mOQP = 60.00 S mQPO = 60.00 mSRT = 60.00 R mRTS = 60.00 Q T Since OPQ and STR both have two angles who measure 60, we can say these triangles are similar by the AA Similarity Theorem. This theorem allows us to say this without any more information, simply having two pairs of congruent angles is enough to say the triangles are similar. Using symbols, OPQ STR. Section 7-5: Theorems for Similar Triangles Theorem 7-1: SAS Similarity THM If an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar. X V mUVA = 30.00 mWXY = 30.00 A U Y m VA = 2.50 cm m VA m XY = 7.45 cm m XY = 0.34 m VU = 2.16 cm m VU m XW = 6.45 cm m XW = 0.34 W Since m UVA WXY and the scale factor that compares VA and XY as well as VU and XW are the same, we can say WXY UVA. Theorem 7-2: SSS Similarity THM If the sides of two triangles are in proportion, then the triangles are similar. Section 7-6: Proportional Lengths Theorem 7-3: Triangle Proportionality THM If a line parallel to one side of a triangle intersects the other two sides, it divides those sides proportionally. Corollary: If three parallel lines intersect two transversals, then they divide the transversals proportionally Theorem 7-4: Triangle Angle-Bisector THM If a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the other two sides.