Proving That a Quadrilateral is a Parallelogram

1. Proving That a Quadrilateral is a Parallelogram Lesson 6-3 Lesson Quiz Find the values of the variables for which GHIJ must be a parallelogram. 1. 2. a = 34, b = 26 x = 6, y = 0.75 Determine whether the quadrilateral must be a parallelogram. Explain. 3. 4. 5. Yes; the diagonals bisect each other. Yes; one pair of opposite sides is both congruent and parallel. No; both pairs of opposite sides are not necessarily congruent. 6-4

2. Special Parallelograms Lesson 6-4 Notes A segment bisects an angle if and only if it divides the angle into two congruent angles. 6-4

3. Special Parallelograms Lesson 6-4 Notes Proof: ABCD is a rhombus, so its sides are all congruent. Therefore, ABC ADC by the SSS Postulate. 12 and 34 by CPCTC. Therefore, bisects BAD and BCD by the definition of bisect. You can show similarly that bisects ABC and ADC. 6-4

4. Special Parallelograms Lesson 6-4 Notes In the rhombus above, points B and D are equidistant from A and C. By the Converse of the Perpendicular Bisector Theorem, they are on the perpendicular bisector of segment AC. 6-4

5. Special Parallelograms Lesson 6-4 Notes 6-4

6. Special Parallelograms Lesson 6-4 Notes 1. Rectangle ABCD 1. Given 2. Definition of rectangle 2. ABCD is a  3.  → opp. sides  4. DAB & CBA are right s 4. Definition of rectangle 5. DABCBA 5. All right s are  6. Reflexive POC 7. DABCBA 7. SAS 8. CPCTC 6-4

7. Special Parallelograms Lesson 6-4 Notes 6-4

8. Theorem 6–9 states that each diagonal of a rhombus bisects two angles of the rhombus, so m 1 = 78. Theorem 6-10 states that the diagonals of a rhombus are perpendicular, so m 2 = 90. Because the four angles formed by the diagonals all must have measure 90, 3 and ABD must be complementary. Because m ABD = 78, m 3 = 90 – 78 = 12. Finally, because BC = DC, the Isosceles Triangle Theorem allows you to conclude 1 4. So m 4 = 78. Special Parallelograms Lesson 6-4 Additional Examples Finding Angle Measures Find the measures of the numbered angles in the rhombus. Quick Check 6-4

9. 5x + 11 = 8x + 2 Diagonals of a rectangle are congruent. 11 = 3x + 2 Subtract 5x from each side. 9 = 3x Subtract 2 from each side. 3 = x Divide each side by 3. 8x + 2 = 8(3) + 2 = 26 Substitute. 5x + 11 = 5(3) + 11 = 26 Special Parallelograms Lesson 6-4 Additional Examples Finding Diagonal Length One diagonal of a rectangle has length 8x + 2. The other diagonal has length 5x + 11. Find the length of each diagonal. By Theorem 6-11, the diagonals of a rectangle are congruent. The length of each diagonal is 26. Quick Check 6-4

10. Special Parallelograms Lesson 6-4 Additional Examples Identifying Special Parallelograms The diagonals of ABCD are perpendicular. AB = 16 cm and BC = 8 cm. Can ABCD be a rhombus or rectangle? Explain. Use indirect reasoning to show why ABCD cannot be a rhombus or rectangle. Suppose that ABCD is a parallelogram. Then, because its diagonals are perpendicular, ABCD must be a rhombus by Theorem 6-12. But AB = 16 cm and BC = 8 cm. This contradicts the requirement that the sides of a rhombus are congruent. So ABCD cannot be a rhombus, or even a parallelogram. Quick Check 6-4

11. • By Theorem 6-7, if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. • By Theorem 6-13, if the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. Special Parallelograms Lesson 6-4 Additional Examples Real-World Connection Explain how you could use the properties of diagonals to stake the vertices of a play area shaped like a rhombus. One way to stake a play area shaped like a rhombus would be to cut two pieces of rope of any lengths and join them at their midpoints. Then, position the pieces of rope at right angles to each other, and stake their endpoints. Quick Check 6-4

12. m 1 = 62, m 2 = 62, m 3 = 56 m 1 = 90, m 2 = 20, m 3 = 20, m 4 = 70 Special Parallelograms Lesson 6-4 Lesson Quiz 1. The diagonals of a rectangle have lengths 4 + 2x and 6x – 20. Find x and the length of each diagonal. 6; each diagonal has length 16. Find the measures of the numbered angles in each rhombus. 3. 2. Determine whether the quadrilateral can be a parallelogram. If not, write impossible. Explain. 4. Each diagonal is 15 cm long, and one angle of the quadrilateral has measure 45. 5. The diagonals are congruent, perpendicular, and bisect each other. Impossible; if diagonals of a parallelogram are congruent, the quadrilateral is a rectangle, but a rectangle has four right angles. Yes; if diagonals of a parallelogram are congruent, the quadrilateral is a rectangle, and if diagonals of a parallelogram are perpendicular, the quadrilateral is a rhombus, and a rectangle that is a rhombus is a square. 6-4

13. Special Parallelograms Lesson 6-4 Check Skills You’ll Need (For help, go to Lesson 6-2.) PACE is a parallelogram and mPAC = 109. Complete each of the following. 1.EC = ? 2.EP = ? 3.mCEP = ? 4.PR = ? 5.RE = ? 6.CP = ? 7.mEPA = ? 8.mECA = ? 9. Draw a rhombus that is not a square. Draw a rectangle that is not a square. Explain why each is not a square. Check Skills You’ll Need 6-4

14. Special Parallelograms Lesson 6-4 Check Skills You’ll Need Solutions 1.EC = AP = 4.5 2.EP = AC = 7 3.mCEP = mPAC = 109 4.PR = CR = 4.75 5.RE = AR = 3.5 6.CP = 2RC = 2(4.75) = 9.5 7. Since PACE is a parallelogram, AC || PE . By the same-side Interior Angles Theorem, PAC and EPA are supplementary. So, mEPA = 180 – mPAC = 180 – 109 = 71. 8. From Exercise 7, mEPA = 71. Since PACE is a parallelogram, opposite angles are congruent. So, mECA = mEPA = 71. 9. Answers may vary. Samples given. The rhombus is not a square because it has no right s. The rectangle is not a square because all 4 sides aren’t . 6-4