Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Euclidean geometry wikipedia, lookup

Pythagorean theorem wikipedia, lookup

History of geometry wikipedia, lookup

Rational trigonometry wikipedia, lookup

Integer triangle wikipedia, lookup

Trigonometric functions wikipedia, lookup

Multilateration wikipedia, lookup

Tessellation wikipedia, lookup

Regular polytope wikipedia, lookup

Technical drawing wikipedia, lookup

History of trigonometry wikipedia, lookup

Dessin d'enfant wikipedia, lookup

Golden ratio wikipedia, lookup

Euler angles wikipedia, lookup

Reuleaux triangle wikipedia, lookup

Penrose tiling wikipedia, lookup

Transcript

Math 2 Geometry Based on Elementary Geometry, 3rd ed, by Alexander & Koeberlein 5.2 Similar Triangles and Polygons Informal Definition • When two figures have the same shape, they are said to be similar. Informal Definition • When two figures have the same shape, they are said to be similar. • When two figures have the same shape, and all corresponding parts have equal measures, the two figures are congruent. Symbols • Equal = Symbols • Equal = • Similar ~ Symbols • Equal = • Similar ~ • Congruent Symbols • Equal = • Similar ~ • Congruent • The Congruent symbol is a combination of the symbols for Equal and Similar Intuitive Definition • Two figures are similar if one is an enlargement of the other. Definition Two polygons are similar if and only if two conditions are satisfied: 1. 2. Definition Two polygons are similar if and only if two conditions are satisfied: 1. All pairs of corresponding angles are congruent. 2. Definition Two polygons are similar if and only if two conditions are satisfied: 1. All pairs of corresponding angles are congruent. 2. All pairs of corresponding sides are proportional. Which Figures must be Similar? • Any two isosceles triangles. Which Figures must be Similar? • Any two isosceles triangles. • Any two regular pentagons. Which Figures must be Similar? • Any two isosceles triangles. • Any two regular pentagons. • Any two rectangles. Which Figures must be Similar? • • • • Any two isosceles triangles. Any two regular pentagons. Any two rectangles. Any two squares. Which Figures must be Similar? • • • • • Any two isosceles triangles. Any two regular pentagons. Any two rectangles. Any two squares. Any two rhombuses. Example Given ABC ~DEF, with indicated measures. Find the measures of the remaining parts of each triangle. F C 12 3 53 A 5 B D E Postulate 15 (AAA) If the three angles of one triangle are congruent to the three angles of a second triangles, then the triangles are similar. Corollary 5.2.1 (AA) If two angles of a triangle are congruent to two angles of another triangle, then the triangles are similar. CSSTP • What could these letters stand for? CSSTP Corresponding sides of similar triangles are proportional. Theorem 5.2.2 The lengths of the corresponding altitudes of similar triangles have the same ratio of any pair of corresponding sides. F C A B E F What would be an outline for proving this? Theorem 5.2.3 (SAS~) If an angle of one triangle is congruent to an angle of a second triangle and the pairs of sides that form the angles are proportional, then the triangles are similar. Theorem 5.2.4 (SSS~) If the three sides of one triangle are proportional to the three corresponding sides of a second triangle, then the triangles are similar.