Using efinitions and Theorems in Proofs midpoint divides a segment into 2 segments midpoint divides a segment in half bisector intersects a segments at its midpoint n angle bisector divides an angle into 2 angles n angle bisector divides an angle in half Perpendicular Lines intersect at right angles Right angles measure 90 n altitude is a segment from a vertex to the opposite side of a median is a segment from a vertex to the opposite midpoint of a 1. Given: is the midpoint of Prove: 3. Given: bisects Prove: 2. Given: bisects at X Prove: = L 4. Given: Prove: m = 90 T X R M
5. Given: is the perpendicular bisector of Prove: m = 90 and 8. Given: Prove: is an altitude of SR R S T 6. Given: is a median of RQT Prove: Q 9. Given: O is the midpoint of Prove: is a median of FLY R S T 7. Given: is an altitude of RQT Prove: is a measures 90 Q 10. Given: FYO LYO Prove: is an angle bisector R S T
Some more efinitions with a right angle is a right with two sides is an isosceles triangle 11. Given: Prove: is isosceles K S 12. Given: Prove: HJL is a right triangle
Proofs Using the Properties of quality Postulate statement that is an obvious truth and does not require a proof. Theorem true statement that has been proven. The ddition and Subtraction Postulates of quality When equal quantities are added (or subtracted) from equal quantities, the sums (or differences) are equal. Write out the two equalities in two separate statements, then add or subtract them in a third statement. The reason is the ddition Postulate or Subtraction Postulate. We also have multiplication and division postulates which work the same way! 15. Given: m = 42 and mf = 48 Prove: and F are complementary 17. Given: 3x + 4 = 19 2x + 10 = 20 Prove: 5x + 14 = 39 16. Given: mlp = 107 and mmf = 73 Prove: LP and MF are supplementary 18. Given: x = 12 3y = 14 Prove: x + 3y = 26
The Reflexive Postulate of quality/ongruence ny quantity is equal to itself ny figure is congruent to itself Obvious but we need to use this when adding or subtracting the same quantity from both sides of an equation. It s common sense, but we will use this one a lot. The Substitution Property of quality quantity can be substituted for any equal quantity Now things are going to get interesting!!! 19. Given: 2x + 12 = 20 Prove: x = 4 21. Given: mpq = 84 bisects PQ Prove: mps = 42 20. Given: 6x - 3 = x + 27 Prove: x = 6 22. Given: = Prove: + = +
23. Given: G is the midpoint of H is the midpoint of Prove: F G H I 25. Given: bisects RST mrsx = 3x 2 mrst = 9x 76 Prove: mrst = 140 T S X R 24. Given: is the midpoint of is the midpoint of Prove: 26. Given:, bisects bisects Prove: F (you can use the postulate all right angles are congruent ) F F
27. Given:, X is the midpoint of, Z is the midpoint of, = 10, = 12, and ZX = 4. Prove: the perimeter of XZ = 15 Z X 29. Given: Y is the midpoint of and X is the midpoint of Prove: WX = ¼ WZ (you can use the reason simplify when doing basic algebraic steps like combining like terms or multiplying two coefficients) 28.Given: R, R Prove: perimeter of R = perimeter of R W X Y Z R
The Partition Postulate The whole equals the sum of the parts + + = + = m + m = m Partition is often combined with the addition or subtraction postulates. 30. Given: Prove: + = 32. Given: mgfh = 40 and mfh = 50 Prove: FG is a right angle F G H T S 31. Given: QR = ST Prove: QS = RT Q R 33. Given: msog = 37 mso = 98 Prove: mog = 135
34. Given:, Prove: 37. hallenge Given: MO JL, KN JL, KN MO Prove: The area of JLM equals the area of JLK M 35. Given:, Prove: J N O L K 36. Given: m = 60, m = 30 Prove: is a right triangle 38. hallenge Given: I VR Prove: R VI R I V
Proofs Using Theorems Vertical angles are congruent Linear pairs are supplementary ngle sum theorem xterior angle theorem Isosceles triangle theorem 39.Given:, 41. Given: intersects at Prove: Prove: 40. Given: and intersect at point O. Prove: O O (use the theorem supplements of congruent angles are congruent O 42.Given: O and OF are complementary Prove: O and O are complementary O F
uuur 43. Given: bisects F, F Prove: is a right angle 45. Given:, Prove: F 44. Given: RT ST Prove: 1 3 T 46. Given:,, and Prove: m + m + m = 180 1 R 2 3 S
47. Given:,,, Prove: m1 + m2 = m4 + m5 4 1 49. Given: with extended to, = Prove: m1 = m2 + m4 4 5 6 3 2 1 2 3 48. Given: JM JK, KM bisects JKL and JML Prove: KLM is isosceles M 50. Given, F, m = 50, and mf = 40 Prove: J L K F