6-5 P R OV I N G R H O M B U S E S, R E C TA N G L E S, A N D S Q UA R E S

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1 6-5 P R OV I N G R H O M B U S E S, R E C TA N G L E S, A N D S Q UA R E S

2 Workbook page 261, number 13 Given: ABCD is a rectangle Prove: EDC ECD A D E B C Statements Reasons 1) ABCD is a rectangle 1) Given 2) AC DB 2) In a rectangle, diagonals are congruent 3) ABCD is a parallelogram 3) All rectangles are parallelograms 5) DE EB; AE EC 5) In a parallelogram, diagonals bisect each other 6) AC = DB; DE = EB; AE = EC 6) Definition of congruent segments 7) AE + EC = AC; DE + EB = DB 7) Segment Addition Postulate 8) AE + EC = DE + EB 8) Substitution 8) EC + EC = DE + DE 8) Substitution 9) 2EC = 2DE 9) CLT 10) EC = DE 10) Division Property of Equality 11) EC DE 11) Definition of congruent segments 12) EDC ECD 12) Isosceles Triangle Theorem

3 Workbook page 261, number 13 Given: ABCD is a rectangle Prove: EDC ECD A D E B C Statements Reasons 1) ABCD is a rectangle 1) Given 2) AC DB 2) In a rectangle, diagonals are congruent 3) ABCD is a parallelogram 3) All rectangles are parallelograms 4) AD BC 4) In a parallelogram, opposite sides congruent 5) DC DC 5) Reflexive Property of Congruence 6) ACD BDC 6) SSS Postulate 7) EDC ECD 7) CPCTC

4 OBJECTIVE TO CLASSIFY AND USE THE PROPERTIES OF SPECIAL TYPES OF PARALLELOGRAMS

5 KEY CONCEPTS To prove a parallelogram is a rhombus: Prove that one pair of consecutive sides are congruent. Prove the diagonals are perpendicular. Prove one diagonal bisects a pair of opposite angles.

6 KEY CONCEPTS To prove a parallelogram is a rectangle: Prove that one of the angles is a right angle. Prove the diagonals are congruent. AC BD. To prove a square, you must be able to prove parallelogram, rectangle, and rhombus. A D B C

7 CLASSWORK Can you conclude that the parallelogram is a rhombus, a rectangle, or a square? Explain

8 CLASS WORK For what 4. value of x is the parallelogram a rhombus? 5.

9 CLASS WORK For what value 6. of x is the parallelogram a rectangle? 7.

10 PROOF OF THEOREM F G Statements Given: EFGH is a parallelogram. EG HF. Prove: EFGH is a rectangle. Reasons E H EFGH is a parallelogram.; EG HF EF HG EH EH FEH GHE FEH GHE FEH and GHE are supplementary FEH and GHE are right angles FEH FGH; GFE GHE FGH and GFE are right angles EFGH is a rectangle Given In a parallelogram, opposite sides congruent Reflexive Property of Congruence SSS Postulate CPCTC In a parallelogram, consecutive angles are suppl. Angles that are congruent and supplementary are right angles In a parallelogram, opposite angles are congruent If an angle is congruent to a right angle it is a right angle Definition of Rectangle

11 EXIT PROBLEMS Determine whether the parallelogram is a rhombus, a rectangle, or a square. Give the most precise description in each case. 8. A parallelogram has perpendicular diagonals and angle measures of 45, 135, 45, and A parallelogram has perpendicular diagonals and angle measures that are all A parallelogram has congruent diagonals.

12 LEARNING RUBRIC Got It: Completes general proofs and uses proof to prove special parallelograms Almost There: Uses formulas to prove special parallelograms on the coordinate plane Moving Forward: Applies the properties of special parallelograms to find or check given values of variables that prove special parallelograms Getting Started: Identifies correctly marked diagrams that prove special parallelograms

13 HOMEWORK 6-4 Workbook PS Page Workbook AP Page 267