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Transcript
Geometry Chapter 4 Notes 4.1 Triangles and Angles A triangle is a figure formed by three segments joining three noncollinear points. Names of Triangles Classified by Sides Equilateral - 3 Congruent sides Isosceles - At lease 2 congruent sides Scalene - No congruent sides Classified by Angles Acute - 3 acute angles Equiangular - 3 congruent angles Right - 1 right Angle Obtuse - 1 obtuse angle Labeling a triangle Each of the 3 points jointing the sides of a triangle is a vertex In a triangle, two sides sharing a common vertex are adjacent sides In a right triangle the sides that form the right angle are the legs and the side opposite the right angle is the hypotenuse. Leg hypotenuse Leg In an isosceles triangle, the congruent sides are the legs and the third side is the base. Leg Leg Base Theorem 4.1 – Triangle Sum Theorem The sum of the measures of the interior angles of a triangle is 180° m A + m B + m C = 180° B A C Theorem 4.2 – Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of B the two nonadjacent interior angles. m 1 = m A + m B 1 A Corollary A corollary to a theorem is a statement that can be proved easily using the theorem Corollary to the Triangle Sum Theorem The acute angles of a right triangle are complementary m A + m B = 90° A B 4.2 Congruence and Triangles Definition – When two figures are congruent, there corresponding angles are congruent and their corresponding sides are congruent Theorem 4.3 – Third angle Theorem If 2 angles of one triangle are congruent to 2 angles of another triangle, then the 3rd angles are also congruent B E A D & B E, then C F C A D F Theorem 4.4 – Properties of Congruent Triangles 4.3 Reflexive Every triangle is congruent to itself Symmetric If ABC DEF, then DEF ABC Transitive If ABC DEF & DEF JKL, then ABC JKL Proving Triangles are Congruent (SSS and SAS) Postulate 19 – Side-Side-Side (SSS) Congruence Postulate If 3 sides of one triangle are congruent to 3 sides of a another triangle, then the 2 triangles are congruent Side Side Side BC AC BC AC, and BC AC, then ABC DEF E B A C D F Postulate 20 – Side-Angle-Side (SAS) Congruence Postulate If 2 sides and the included angle of one triangle are congruent to 2 sides and the included angle of a another triangle, then the 2 triangles are congruent Side BC AC Angle A D , and Side BC AC, then ABC DEF 4.4 B E C A F D Proving Triangles are Congruent (ASA and AAS) Postulate 21 – Angle-Side-Angle (ASA) Congruence Postulate If 2 angles and the included side of one triangle are congruent to 2 angles and the included side of a another triangle, then the 2 triangles are congruent Angle A D Side BC AC, and Angle A D , then ABC DEF B A E C D F Theorem 4.5 – Angle-Angle-Side (AAS) Congruence Postulate If 2 angles and a nonincluded side of one triangle are congruent to 2 angles and the corresponding nonincluded side of a another triangle, then the 2 triangles are congruent Angle A D Angle A D, and Side BC AC, then ABC DEF 4.5 Using Congruent Triangles B A E C D F 4.6 Isosceles, Equilateral, and Right Triangles Theorem 4.6 – Base Angles Theorem If 2 sides of a triangle are congruent, then the angles opposite them are congruent. B If AB AC, then B C A C Theorem 4.7 – Converse of the Base Angles Theorem If 2 angles of a triangle are congruent, then the sides opposite them are congruent. If B C, then AB AC B A C Corollary to Theorem 4.5 If a triangle is equilateral, then it is equiangular Corollary to Theorem 4.6 If a triangle is equiangular, then it is equilateral, Theorem 4.8 – Hypotenuse-Leg (HL)Congruence Theorem If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a 2nd triangle, then the two triangles are congruent. If BC AC & AC DF, then ABC DEF 4.7 Triangles and Coordinate Proof