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Summary of Theorems

Summary of Theorems. Postulate 1 (Line Postulate) Through two given points, one and only one can be drawn through them. A. B. Postulate 2 (Unique Intersection Postulate) Two distinct lines can have one and only one point in common. The common point is called the point of intersection. r.

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Summary of Theorems

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  1. Summary of Theorems

  2. Postulate 1 (Line Postulate) Through two given points, one and only one can be drawn through them. A B

  3. Postulate 2 (Unique Intersection Postulate) Two distinct lines can have one and only one point in common. The common point is called the point of intersection. r P t

  4. Postulate 3 (Right Angle Postulate) All right angles are equal.

  5. B A C • Postulate 4 (Whole-part Postulate) The whole is equal to the sum of its parts and is greater than any of them. Ac = BC + BC

  6. Postulate 5 (Angle Addition Postulate) If AOC is composed of two adjacent angles, AOB and BOC, then AOC=AOB+BOC,AOC-AOB=BOC and AOC-BOC=AOB. B C d c A O

  7. Theorem 3.1 (Linear Pair Theorem) If two adjacent angles a and b form a straight angle, they are supplementary. B b a A C O

  8. Theorem 3.2 (Supplements Theorem) Supplements of the same or of equal angles are equal. d a b c

  9. Theorem 3.3 (Complements Theorem) Complements of the same or of equal angles are equal. d c b a

  10. Theorem 3.4 (Vertical Angles Theorem) When two lines intersect, any two vertical angles formed are equal. m b c a d n

  11. Theorem 3.5 (Equal Linear Pair) If two lines intersect forming a linear pair of equal angles, then the lines are perpendicular. C 2 1 A B

  12. The SAS Congruence Postulate (SAS) Two triangles are congruent if two sides and the included angle of one triangle are equal respectively, to two sides and the included angle of the other. A x B C Z Y

  13. Theorem 4.1 (Either Isosceles Triangle Theorem or Angles Opposite Equal Sides) Base angles of an isosceles triangle are equal. A B C D

  14. Theorem 4.2 (Bisector of Vertex Angle) The bisector of the vertex angle of an isosceles triangle bisects the base.

  15. Theorem 4.3 ( L.L. ) Two right triangles are congruent if the legs of one triangle are equal to the legs of the other. x A Y Z C B

  16. Theorem 4.4 (Exterior Angle Theorem 1) An exterior angle of a triangle is greater than either non- adjacent interior angle. A F 1 M 2 3 4 B C D

  17. The ASA Congruence Postulate Two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of the other. x A Y Z B C

  18. Theorem 4.5 ( L.A. ) Two right triangles are congruent if a leg and an adjacent acute angle of one triangle are equal respectively to the corresponding parts of the other. A x B Y Z C

  19. The SSS Postulate Two triangles are congruent if the three sides of one are equal to the corresponding sides of the other. A x B Y Z C

  20. Corollary 4.6 An equilateral triangle is equiangular. A B C

  21. Theorem 4.6 In a circle or in equal circles equal central angles subtend equal chords and conversely. D Q y O

  22. Theorem 4.7 (Angle Opposite Longer Side) In a triangle, if one side is longer than the other, then the angle opposite the longer side is greater. A x C B

  23. Theorem 5.1 (Perpendicular Bisector Theorem) Any point on the perpendicular bisector of a segment is equidistant from the ends of the segment. P B A M m

  24. Theorem 5.2 (Converse of the Perpendicular Bisector Theorem) Any point equidistant from the ends of a segment lies on the perpendicular bisector of the segment. P 1 2 A B M

  25. The Parallel Postulate Through a point on P not on a line m, one and only one line can be drawn parallel to m. P M

  26. Theorem 5.3 If a line intersects one of two parallel lines, it intersects the other also. T P N M

  27. Theorem 5.4 ( AIP ) If two lines cut by a transversal form equal alternate interior angles, then the two lines are parallel. A 1 P B 2

  28. Corollary 5.5 ( CAP ) If two lines m and n are cut by a transversal t so that corresponding angles formed are equal, then m is parallel to n. t 1 m 3 2 n

  29. Corollary 5.6 (AEP) If two lines m and n are cut by a transversal t such that alternate exterior angles are equal, then m is parallel to n. t 1 m 3 2 n

  30. Corollary 5.7 (ISP) If two lines m and n are cut by a transversal t such that interior angles on the same side of a transversal are supplementary, then m is parallel to n. t 1 m 3 2 n

  31. Corollary 5.8 (ESP): If two lines m and n are cut by a transversal t so that exterior angles on the same side of a transversal are supplementary then m is parallel to n. t 1 m 3 n 2

  32. Theorem 5.9 (lines perpendicular to the same line): If two lines are perpendicular to the same line, then they are parallel to each other. n s g h m

  33. Theorem 5.10 (PAI or alternate interior angles, m is parallel to n) If two parallel lines are cut by a transversal, then alternate interior angles are equal. t R A m 1 2 n B

  34. Corollary 5.11 • If two parallel lines are cut by a transversal, then • Corresponding angles are equal. (corr. Angles, m is parallel to n) • Alternate exterior angles are equal. (alt. ext. angles, m is parallel to n) • Interior angles on the same side of the transversal are supplementary. (int. angles, m is parallel to n) • Exterior angles on the same side of the transversal are supplementary. (ext. angles, m is parallel to n) t 1 m 2 3 4 n 5

  35. Theorem 5.12 (Angle sum of a triangle) The sum of the angles of a triangle equals a straight angle. x A Y 1 3 2 1 3 B C

  36. Theorem 5.13 (Exterior Angle Theorem 2) Exterior angle of a triangle equals the sum of the two remote or angles. A x 2 3 1 B D C

  37. Theorem 5.14 (SAA): • If a side and two angles of a triangle are equal to the corresponding parts of another, then the two triangles are congruent x A Y Z C B

  38. Corollary 5.15 (Hy A) • If the hypotenuse and an acute angle of a right triangle are equal to the corresponding parts of another, then the two right triangles are congruent. A x Y Z B C

  39. Theorem 5.16 (Hy L): • If the hypotenuse and a leg of one right triangle are equal to the corresponding parts of another right triangle, then the two are congruent. x A Y Z B C

  40. Theorem 5.17 (Angle Bisector Theorem): • Any point on the bisector of an angle is equidistant to the sides of the angle. A C Y x B D

  41. Theorem 5.18 (Converse, Angle Bisector Theorem): • Any point equidistant from the sides of an angle lies on the bisector of the angle. A C Y x B D

  42. Theorem 5.19: • If two angles of a triangle are equal, then the triangle is isosceles. A 3 4 1 2 C B D

  43. Theorem 5.19 (Sides Opposite Equal Angles): • If two angles of a triangle are equal, then the sides opposite them are equal.

  44. Corollary 5.20: • An equiangular triangle is equilateral. A B C

  45. Theorem 5.21 (Side opposite Greater Angle): • In a triangle, if one angle is greater than another, then the side opposite the greater angle is greater. A C B

  46. Theorem 5.22 (sum of two sides greater than third side): • The sum of any two sides of a triangle is greater than the third side. X 1 A b c 2 B C a

  47. Corollary 5.23 (Difference of two sides less than third side): • The difference between any two sides is less than the third side.

  48. Theorem 6.1: • In a parallelogram, any two consecutive angles are supplementary and opposite angles are equal. A B C D

  49. Theorem 6.2: • A diagonal divides a parallelogram into two congruent triangles. A B 1 4 3 2 D C

  50. Corollary 6.3: • Opposite sides of a parallelogram are equal. A B 1 3 4 2 D C

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