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Transcript
Theorem Pythagorean Theorem c a b a² + b² = c² p. 20 Theorem Distance Formula A( x1 , y1 ) B ( x2 , y 2 ) AB ( x2 x1 ) ( y2 y1 ) 2 2 p. 19 Postulate Segment Addition Postulate A C B If B is between A and C, then AB + BC = AC. p. 18 Postulate Angle Addition Postulate A P B C If P is in the interior of ABC, then mABP + mPBC = mABC p. 27 Definition midpoint M C A The midpoint is a point that divides or bisects a segment into two equal segments. If M is a midpoint, then AM = MC. p. 34 Definition segment bisector k M C A A segment bisector is a line, ray, segment or plane that intersects a segment at its midpoint. p. 34 Definition angle bisector 1 2 1 2 An angle bisector is a ray that divides an angle into two congruent adjacent angles. p. 36 Theorem Midpoint Formula M A( x1 , y1 ) B( x2 , y2 ) x1 x2 y1 y2 M , 2 2 p. 35 Definition complementary angles 1 2 m1 + m2 = 90 A pair of angles whose sum is 90° are complementary. p. 46 Definition supplementary angles 1 2 m1 + m2 = 180 A pair of angles whose sum is 180° are supplementary. p. 46 Definition right angle 90° An angle whose measure is 90° is a right angle. p. 28 Definition perpendicular lines Two lines are called perpendicular if they intersect to form a right angle. p. 79 Property Reflexive B BD BD A D C For any real number, a = a. p. 96 Property Transitive If AB = CD and CD = EF, then AB = EF. . A .B . C .D . .F E If a = b and b = c, then a = c. p. 96 Property Addition Property of Equality If AB = CD, then AC = BD. . . . . A B C D If a = b, then a + c = b + c. p. 96 Property Subtraction Property of Equality If AC = BD, then AB = CD. . . . . A B C D If a = b, then a c = b c. p. 96 Property Substitution Example: If AB = 5 + x and x = 3, then AB = 8. If a = b, then a can be substituted for b in any equation or expression. p. 96 Theorem Right Angle Congruence Theorem 1 2 1 2 All right angles are congruent. p. 110 Theorem Congruent Supplements Theorem If m1 + m2 = 180 and m2 + m3 = 180, then 1 3. 2 1 3 Two angles supplementary to the same angle (or ’s) are congruent. p. 111 Theorem Congruent Complements Theorem If m1 + m2 = 90 and m2 + m3 = 90, then 1 3. 1 2 3 Two angles complementary to the same angle (or ’s) are congruent. p. 111 Postulate Linear Pair Postulate m1 + m2 = 180 1 2 If two angles form a linear pair, then they are supplementary. p. 111 Theorem Vertical Angles Theorem 1 2 and 3 4 1 3 4 2 Vertical angles are congruent. p. 112 Theorem Linear Pair of s h g h g If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. p. 137 Postulate Corresponding Angles Postulate 1 2 1 2 If two parallel lines are cut by a transversal, then corresponding ’s are . p. 143 Theorem Alternate Interior Angles Theorem 1 2 1 2 If two parallel lines are cut by a transversal, then alt. int. ’s are . p. 143 Theorem Alternate Exterior Angles Theorem 1 2 1 2 If two parallel lines are cut by a transversal, then alt. ext. ’s are . p. 143 Theorem Consecutive Interior Angles Theorem m1 + m2 = 180 1 2 If two parallel lines are cut by a transversal, then consecutive int. ’s are supplementary. p. 143 Theorem Perpendicular Transversal Theorem jm j k m If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. p. 143 Theorem Two Lines Perpendicular to Same Line j // m j m k In a plane, two lines perpendicular to the same line are parallel to each other. p. 157 Theorem Two Lines Parallel to the Same Line m // n m k n If two lines are parallel to the same line, then they are parallel to each other. p. 157 Theorem Triangle Sum Theorem B A C mA + mB + mC = 180 The sum of the measures of the interior angles of a triangle is 180°. p. 196 Theorem Exterior Angle Theorem B A 1 m1 = mA + mB The measure of an exterior angle of a triangle is equal to the sum of the two remote interior angles. p. 197 Theorem Third Angles Theorem B E A If A D and B E, then C F C D F If two angles of one are to two angles of another , the third angles are . p. 203 Postulate SSS Side-Side-Side Congruence B A E C D F If AB DE , BC EF , and CA FD , then ABC DEF. If three sides of one are to three sides of another , then the ’s are . p. 212 Postulate SAS Side-Angle-Side Congruence B A E C D F If AB DE , CA FD and A D, then ABC DEF. If two sides of one are to two sides of another , and the included s are , then the ’s are . p. 213 Theorem Perpendicular/Right Theorem (Meyers Theorem) j m 1 2 k If j k and m k, then 1 2. Perpendicular lines form right s. p. 157 Postulate ASA Angle-Side-Angle Congruence B A E C D F If A D, C F and CA FD , then ABC DEF. If two s of one are to two s of another , and the included sides are , then the ’s are . p. 220 Postulate AAS Angle-Angle-Side Congruence B A E C D F If A D, C F and BC EF , then ABC DEF. If two s of one and a non-included side are to two s of another and the corresponding non-included side, then the ’s are . p. 220 Theorem Base Angles Theorem If two sides of a are , then the s opposite those sides are . p. 236 Theorem Base Angles Converse Theorem If two s of a are , then the sides opposite those s are . p. 236 Theorem Hypotenuse-Leg Theorem H-L A B D C E F If the hypotenuse and a leg of one right are to a hyp. and a leg of another rt. , the two s are . p. 238 Theorem Perpendicular Bisector Theorem k AC = BC C B A P If a point is on the bisector of a segment, then it is equidistant from the endpoints of that segment. p. 265 Theorem Angle Bisector Theorem AP = CP A P B C If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. p. 266 Theorem Circumcenter The perpendicular bisectors of a triangle intersect in a point that is equidistant from the vertices of the triangle. p. 273 Theorem Incenter The angle bisectors of a triangle intersect in a point that is equidistant from the sides of the triangle. p. 274 Theorem Centroid B E A P F D C The medians of a triangle (E, D, and F are midpoints) intersect in a point called a centroid. AP = 2/3 AD, BP = 2/3 BF, CP = 2/3 CE p. 279 Theorem Orthocenter The altitudes of a triangle intersect in a point of concurrency called an orthocenter. p. 281