A proportion is an equation that two ratios are equal. For example, See the diagram. a. Find the ratio of AE to BE.
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1 Section 1: Ratio and Proportion The ratio of a to b means a/b. For example, the ratio of 4 to 6 (or 4:6) is ; the ratio of x to y (or x:y) is proportion is an equation that two ratios are equal. For example, the proportion of a:b=c:d is same as xample See the diagram. a. Find the ratio of to. b. Find the ratio of the largest angle of triangle to the smallest angle of triangle. 2. rectangular field has a length of one kilometer 5x and a width of 300 meters. Find the ratio of the length to the width. 3. telephone pole 7 meters is divided into the ratio of 3:2. Find the lengths. Section 1: Ratio and Proportion is a parallelogram. Find each ratio. 1. : 2. : 6 3. m!:m! 4. :perimeter of : x=2 and y=3. Write each ratio in simplest form. 5. x to y 6. 6x 2 to 12xy 7. Write each algebraic ratio in simplest form
2 Section 2: Properties of Proportions Properties of Proportions 1. is equivalent to 2. If, then xample NOT: a & d are called extremes and b & c are called means. 1a is called the means-extremes multiplication property Use the proportion 1. 5a= to complete each statement Section 2: Properties of Proportions 1. If, then 2x= 2. If 2x=3y, then 3. If, then 4. If, then In the figure, 5. If =2, =6, and =3, then = 6. If =10, =8, and =, then =
3 Section 3: Similar Polygons Two polygons are similar (denoted ~) if their vertices can be paired so that: orresponding angles are congruent orresponding sides are in proportion. Let us say that polygon ~ polygon PQRST P T Q S R From the definition of similar polygons, we have: (complete the list) (1)!"!P,! "!,! "!,! "!, and! "!. (2) Section 3: Similar Polygons xample 1. Quadrilateral ~ quadrilateral. a. find their scale factor b. the values of x, y, and z c. the ratio of the perimeter 2. Quadrilateral FGH ~ quadrilateral F G H a. find their scale factor b. the values of x, y, and z c. the ratio of the perimeter
4 Section 3: Similar Polygons 1. Quadrilateral ~ quadrilateral FGH. a. m!= b. m!g= c. m!= d. If m!=110, then m!h= k 2 e. The scale factor is f. H= g. = h. = Section 4: Postulate for Similar Triangles efinition Two triangles are similar if and only if, 1. all corresponding angle of two triangles are congruent, and 2. all proportion of corresponding sides of triangle are equal. Postulate 15 ( Similarity Triangle) If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
5 Section 4: Postulate for Similar Triangles xample 1. Given:. Prove: #O~#O. (Provide the reasons for each step.) 1. 2.!"!;!"! 3. #O~#O 2. Given: $F; RH$H;!1"!2. Prove: HR F= H (Provide the reasons for each step.) 1. $F; RH$H;!1"!2 2.!RH"!F 3. #RH~#F HR F= H Section 4: Postulate for Similar Triangles Tell whether each triangles are similar 1. or not. Find the value of x a. #~ b. y= 4. c., so x=
6 Section 5: Theorems for Similar Triangles Theorem (SS Similarity Theorem) 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. F Theorem (SSS Similarity Theorem) If the sides of two triangles are in proportion, then the triangles are similar. F Section 5: Theorems for Similar Triangles xample 1. The measures of the sides of # are 4, 5, and 7, and the measures of the sides of # XYZ are 16, 20, and 28. re the triangle similar? If so, justify. If not, why not? 2. In #, =2, =5, and =6. In #XYZ, XY=2.5, YZ=2, and XZ=3. Is # ~ #XYZ? If so, justify. If not, why not? X P Q 3. If #XYQ ~ #XZP, does it follow that #XPQ~#XZY? Y Z
7 Section 5: Theorems for Similar Triangles PRTI Name similar triangles and state the postulate or theorem that justifies your answer If #~#F, does the segment correspond to the segment? 2. oes the segment correspond to segment F? oes the segment correspond to segment F? Given:!"! Prove: #%# Section 6: Proportional Lengths Theorem (Triangle Proportionality Theorem) If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally. Given: #RST; PQ RS Prove: Statements Reasons
8 Section 6: Proportional Lengths orollary If three parallel lines intersect two transversals, then they divide the transversals proportionally. Given: Prove: Theorem (Triangle ngle-isector Theorem) If a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the other two sides. Given: #F; G bisects!f Prove: Section 6: Proportional Lengths 1. a. b. If =3, =6, and =3.5, then = c. If =12, =8, and =6, then = 2. a. If a=2, b=3, and c=5, then d= b. If a=4, b=8, c=5, then c+d=
9 Section 6: Proportional Lengths 1. True or false? a. b. c. d. e. f. 2. True or false? a. b. c. d. Find the value of x
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