* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download 4 - Wsfcs
Survey
Document related concepts
History of geometry wikipedia , lookup
Penrose tiling wikipedia , lookup
Dessin d'enfant wikipedia , lookup
Golden ratio wikipedia , lookup
Technical drawing wikipedia , lookup
Multilateration wikipedia , lookup
Apollonian network wikipedia , lookup
Euler angles wikipedia , lookup
Rational trigonometry wikipedia , lookup
Reuleaux triangle wikipedia , lookup
Trigonometric functions wikipedia , lookup
History of trigonometry wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Transcript
4.1 Triangles and Angles Triangle: a figure formed by three segments joining three noncollinear points. It can be classified by its sides and by its angles. Classification by Sides Equilateral Isosceles 3 congruent sides at least 2 congruent sides no congruent sides Classification by Angles Acute Equiangular 3 acute angles Scalene 3 congruent angles Right Obtuse 1 right angle 1 obtuse angle Vertex: each of three points joining the sides of a triangle. A, B, and C are vertices of triangle ABC. A B C Adjacent sides: two sides sharing a common vertex. AB and BC share vertex B. Right and Isosceles Triangles Right Triangle Isosceles Triangle hypotenuse leg leg leg leg base Triangle Sum Theorem: The sum of the measures of the interior angles of a triangle is 180*. m<A + m<B + m<C = 180* A B C Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. m<1 = m<A + m<B A B C Corollary to the Triangle Sum Theorem: The acute angles of a right triangle are complementary. m<A + m<B = 90* A C B 4.2 Congruence and Triangles Congruent: two geometric figures that have exactly the same size and shape. When two figures are congruent, there is a correspondence between their angles and sides such that corresponding sides are congruent and corresponding angles are congruent. ABC = PQR A B Corresponding Angles <A = <P <B = <Q <C = <R P C R Corresponding Sides AB = PQ BC = QR CA = RP Q Third Angle Theorem: If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent. If <A = <P and <B = <Q, then <C = <R Reflexive Property of Congruent Triangles: Every triangle is congruent to itself. Symmetric Property of Congruent Triangles: If ABC = DEF, then DEF = ABC. Transitive Property of Congruent Triangles: If ABC = DEF and DEF = JKL, then ABC = JKL 4.3 Proving Triangles are Congruent: SSS and SAS SSS Congruence Postulate If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. If Side AB = DE C F Side BC = EF Side CA = FD Then ABC = DEF A B E D SAS Congruence Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. If Side AB = DE C F Angle <B = <E Side BC = EF Then ABC = DEF A B E D Choosing Which Congruence Postulate to Use Name the included angle between the given pair of sides. AB and BC CE and DC AC and BC 4.4 Proving Triangles are Congruent: ASA and AAS ASA Congruence Postulate If two angles and the included side on one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent. If Angle <C = <F C F Side BC = EF Angle <B = <E Then ABC = DEF A B E D AAS Congruence Postulate If two angles and the nonincluded side of one triangle are congruent to two angles and the nonincluded side of a second triangle, then the two triangles are congruent. If Angle <C = <F C F Angle <A = <D Side AB = DE Then ABC = DEF 4.5 Using Congruent Triangles EQ: What does the acronym CPCTC represent? Knowing that all pairs of corresponding parts of congruent triangles are congruent can help one prove congruent parts of triangles. Given: PS = RS, PQ = RQ Prove: <PQS = <RQS Statements Reasons 1. PS = RS 1. Given 2. PQ = RQ 2. Given 3. QS = QS 3. Reflexive 4. PQS = RQS 4. SSS 5. <PQS = <RQS 5. CPCTC Q P R S Once one proves two triangles are congruent, then any pair of congruent parts are congruent by CPCTC: Corresponding Parts of Congruent Triangles are Congruent. These may be corresponding sides or angles. 4.6 Isosceles, Equilateral, and Right Triangles Base Angle Theorem: if two sides of a triangle are congruent, then the angles opposite them are congruent. If AB = AC, then <B = <C Base Angle Converse: if two angles of a triangle are Congruent, then the sides opposite them are congruent. If <B = <C, then AB = AC Corollary: if a triangle is equilateral, then it is equiangular. Corollary: if a triangle is equiangular, then it is equilateral. HL: Hypotenuse-Leg: if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent. In the two right triangles if leg BC = EF and hypotenuse AC = DF then ABC = DEF by HL Triangle may be proved congruent by any of five ways SSS SAS ASA AAS HL 4.7 Triangles and Coordinate Proof Place a 2 unit by 6 unit rectangle in a coordinate plane. A right triangle has legs of 5 and 12 units. Place the triangle in a coordinate plane and then find the length of the hypotenuse. In the diagram ∆MLO = ∆KLO. Find the coordinate of L Write a plan to prove that SQ bisects <PSR. Can the two triangles be proved congruent? If so, which postulate or theorem can be used? 1. 4. 2. 5. 6. 7. 10. 3. 8. 11. 9. 12. 13. Write the SSS Congruence Postulate in your own words. 14. Write the SAS Congruence Postulate in your own words. 15. Write the ASA Congruence Postulate in your own words. 16. Write the AAS Congruence Theorem in your own words. Find the values of the missing angles or sides. 17. 18. 19. 20. 21.