Geometry Definitions, Postulates, and Theorems

Transcription

1 Geometry efinitions, Postulates, and Theorems hapter : Similarity Section.1: Ratios, Proportions, and the Geometric ean Standards: Prepare for 8.0 Students know, derive, and solve problems involving the perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures.. Simplify the following ratios. Ratio Two quantities that are measured in the same units. They are usually epressed in simplest form. Two ratios that have the same simplified form are called equivalent ratios. Ratios can be written three ways: b a, The denominator, b, cannot be zero. a : b, or a to b. 5ft. a) 4m : m b) 20in. 4lb. c) 12oz.. ind the ratio of the width to the length of the rectangle. cm m. The perimeter of a rectangle is 8 feet. The ratio of the length to the width is 8 : 1. ind the length and the width.. The perimeter of the isosceles triangle is 5 in. The ratio of :N is 5:4. ind the lengths of the sides and the base of the triangle. N. The measures of the angles of a triangle are in the etended ratio of 3:4:8. ind the measures of the angles of the triangle. (over)

2 Proportion n equation that states that two ratios are equal. a b c d tremes The values on the top left and bottom right in a proportion. eans The values on the top right and bottom left in a proportion. ross Product Property In a proportion, the product of the etremes equals the product of the means. a c I, b d THN ad bc. Solve the proportions. a) 9 5 b) 14 4 c) y 3 y 7 14 X. The ratio of : is 2:9. ind. 27 Geometric ean I a & b are positive numbers, THN the positive number,, represents the geometric mean in the proportion: a b. ind the geometric mean of the two numbers. a) 4 and 9 b) 5 and 25 c) 7 and 20

3 Section.2: Use Proportions to Solve Geometry Problems Standards: 12.0 Students find and use measures of sides and of interior and eterior angles of triangles and polygons to classify figures and solve problems. dditional Properties of Proportions I a c a b a c a + b c + d, THN I, THN b d c d b d b d Reciprocal Property If two ratios are equal, then their reciprocals are also equal. I a c b d, THN b d a c Scale rawing drawing that is the same shape as the object it represents. Scale ratio that describes how the dimensions in the drawing are related to the actual dimensions of the object.. omplete the statements: a 3 a) If, b 4 a then 3 p 5 b) If, q 8 p + q then q. ecide whether the statement is true or false. m 4 a) If, n 5 then n 4 m 5 d 3 b) If, g 4 then d g g 1 4. Use the diagram and the given information to find the unknown length., a) Given Q Q, find Q. b) Given, find. N P N 4 5 P Q

4 Section.3: Use Similar Polygons Standards: 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles. 8.0 Students know, derive, and solve problems involving the perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures Students determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures and solids. Similar Polygons Two or more polygons where: 1. The corresponding angles are congruent and ~GH 2. orresponding sides are proportional. *Similar is not the same as congruent. (shrunk vs. blown up) G H. Given JK ~ PQR. ist all pairs of congruent angles and write the ratios of the corresponding side lengths in a statement of proportionality. Scale actor The ratio of the corresponding sides of two similar polygons.. etermine whether the polygons are similar. If they are, write a similarity statement and find the scale factor. K ***Theorem.1 Perimeters of Similar Polygons If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths.. In the diagram to the right, ~GHJK a) ind the scale factor of to GHJK b) ind the value of. 20 c) ind the perimeter of. 18 K G 9 H 12 J. In the diagram, orresponding engths in Similar Polygons If two polygons are similar, then the ratio of any two corresponding lengths in the polygons is equal to the scale factor of the similar polygons. TPR ~ XPZ. ind the length of the altitude PS. T S R P 20 X 8 Y 8 Z

5 Section.4: Prove Triangles Similar by Standards: 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles Students find and use measures of sides and of interior and eterior angles of triangles and polygons to classify figures and solve problems. ngle-ngle () Similarity Postulate I two angles of one triangle are congruent to two angles of another triangle, THN the two triangles are similar. K Y J X Z J. etermine whether the triangles are similar. If they are, write a similarity statement. X a) b) 44 0 W 52 0 Y V Z Shared angle. In the diagram to the right, N ~ PQN. a. Write the proportionality statement. 33 b. ind m 20 0 N m P Q P c. ind N. Use a proportion to find the value of in the diagram. 4 5 (over)

6 Section.4 Side I Side II Perimeter I Perimeter II edian I edian II ngle isector I ngle isector II ltitude I ltitude II Scale actor Side I Side II Perimeter I Perimeter II iagonal I iagonal II Scale actor

7 Section.5: Prove Triangles Similar by SSS and SS Standards: 4.0 Students prove basic theorems involving congruence and similarity. 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles. ***Theorem.2 Side-Side-Side (SSS) Similarity Theorem I the corresponding side lengths of two triangles are proportional, THN the triangles are similar. *no information about angles is given Reduce ***Theorem.3 Side-ngle-Side (SS) Similarity Theorem I an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, THN the triangles are similar. *one pair of angles are congruent and adjacent sides are proportional Which of the following three triangles are similar? Write a similarity statement. iggest edium Smallest Q R 18 N K P S 5 O (over)

8 . re these two triangles similar? Sides are not proportional, so triangles aren't similar.. re these two triangles similar? Y 4 9 X Z ll sides are proportional so the triangles are similar.. Use the diagram below to complete the statements G 1. m G 2. m G 3. m G G. G 45; alt. int. angles r congruent 50; alt. int. angles r congruent ; vertical angles 7. Name the three pairs of ' s that are similar ~.

9 Section.: Proportions and Similar Triangles Standards: 4.0 Students prove basic theorems involving congruence and similarity. 7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal. ***Theorem.4 Triangle Proportionality Theorem I a line parallel to one side of a triangle intersects the other two sides, THN it divides the two sides proportionally P 9 T 24 ***Theorem.5 onverse of the Triangle Proportionality Theorem I a line divides two sides of a triangle proportionally, THN it is parallel to the third side. Q R S ***Theorem. I three parallel lines intersect two transversals, THN they divide the transversals proportionally. ***Theorem.7 I a ray bisects an angle of a triangle, THN it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides.. p 3 4 (over)

10 X. H True or alse 1. H K K 2. H HK 3. H K 4. H K X. ind and y y

11 Section.7: Perform Similarity Transformations Standards: 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles Students determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures and solids. ilation transformation that stretches or shrinks a figure to create a similar figure. enter of ilation In a dilation, a figure is enlarged or reduced with respect to a fied point. Scale actor of a ilation The ratio of a side length of the image to the corresponding side length of the original figure. oordinate Notation for a ilation You can describe a dilation with respect to the origin with the notation (, y) (k, ky), where k is the scale factor. Reduction If 0 < k < 1, the dilation is a reduction. nlargement If k > 1, the dilation is an enlargement.. raw a dilation of the polygon with the given vertices using the given scale factor k. ( 1,1), (2,2), (3, 1); k 2 y. etermine whether the dilation from igure to igure is a reduction or an enlargement. Then find its scale factor. y