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Geometry Section 6-2C Geometry Section 6-2C Proofs with Special Proofs with Special ParallelgramsParallelgrams

Page 427Page 427Be ready to grade 6-2BBe ready to grade 6-2BQuiz Friday!Quiz Friday!

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Answers for 6-2B

1. No, consecutive angles are not supplementary (1 pt.)

2. Yes, opposite sides are congruent (1 pt.)

3. Yes, the diagonals bisect each other (1 pt.)4. Yes, the diagonals bisect each other (1 pt.)5. No, must have 2 sets of parallel sides (1 pt.)6. No, must have 2 sets of congruent angles (1

pt.)7. Yes, opposite angles are congruent (1 pt.)8. Consecutive angles are supplementary

x = 24, y = 36 (3 pts.)9. Alternate interior angles are congruent

x = 10, y = 7 (3 pts.)10. QPS QRS and PQR RSP (1

pt.)11. QT = TS and PT = TR (or if segments) (1

pt.)

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Answers for 6-2B cont.

12. 10 pts. for the proof. (Order may be different, ask if you have a question.)

1 pt. for a labeled drawing. 2 points for Given and Prove1. AE CE Given2. BE DE Given3. BEC AED Vertical angles are congruent4. AEB CED Vertical angles are congruent5. BEC AED SAS6. AEB CED SAS7. CBD BDA CPCTC8. BC AD Alt. Int. angles theorem9. BCA CAD CPCTC 10. BA CD Alt. Int. angles theorem11. ABCD is a parallel. Def. of parallelogram

28 pts. possible

A

B

D

CE

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GRADE SCALE – 28 POSSIBLE

27.5 – 98% 20.5 – 73%27 – 96% 20 – 71%26.5 – 95% 19.5 – 70%26 – 93% 19 – 68%25.5 – 91% 18.5 – 66%25 – 89% 18 – 64%24.5 – 88% 17.5 – 63%24 – 86% 17 – 61%23.5 – 84% 16.5 – 59%23 – 82% 16 – 57%22.5 – 80% 15.5 – 55%22 – 79% 15 – 54%21.5 – 77% 14.5 – 52%21 – 75% 14 – 50%

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Special Parallelograms:Special Parallelograms:

At the beginning of this chapter, we discussed the quadrilateral “Family Tree”. Since rhombuses,

rectangles and squares are special parallelograms, their diagonals have some unique characteristics. We will use these characteristics to prove that a parallelogram is a rhombus, rectangle or square.

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Explore:Explore:

Pg.428

(Smartboard program)

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Try It:Try It:

Pg.429

Classify this quadrilateral if

Parallelogram

VX and YW bisect each other.

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Try It:Try It:

Pg.429

Classify this quadrilateral if

Rhombus

VX and YW bisect each other and VX YW.

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Try It:Try It:

Pg.429

Classify this quadrilateral if

Rectangle

VX and YW bisect each other and VX YW.

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Try It:Try It:

Pg.429

Classify this quadrilateral if

Square

VX and YW bisect each other, VX YW and

VX YW.

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DEFG is a rectangle Given

DGF and EFG are rt. ‘s Def. of rectangle

DEFG is a parallelogram All rectangles are parallelograms

Def. of right

DG EF Opp. sides of a parallelogram are

DGF and EFG are rt. ‘s

Given: Parallelogram DEFG is a rectangle with diagonals DF and GE.

Prove: DF GE

GF GF Reflexive property

DGF EFG LL Theorem

DF GE CPCTC

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Theorems:Theorems:

Pg.429

Theorems about diagonals of special parallelograms.

A parallelogram is a…

rhombus if and only if it’s diagonals are .

rectangle if and only if it’s diagonals are .

square if and only if it’s diagonals are and.

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Exercises:Exercises:

#1

Pg.430

Classify ABCD if AC and BD bisect each

other.

Parallelogram

THINK! What could each shape look like if the diagonals DON’T bisect each other? Don’t assume too

much!

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Exercises:Exercises:

#2

Pg.430

Classify ABCD if AC and BD bisect each other and are congruent.

Rectangle

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Exercises:Exercises:

#3

Pg.430

Classify ABCD if AC and BD bisect each other

and are perpendicular.

Rhombus

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Exercises:Exercises:

#4

Pg.430

Classify ABCD if AC and BD bisect each other

are perpendicular and congruent.

Square

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Exercises:Exercises:

#13

Pg.431

Find the values of x and y for the quadrilateral. Justify your

answer.

For a rectangle, diagonals must bisect and be congruent.

3x = 15

x = 5

y - 5 = 15

y = 20

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Exercises:Exercises:

#14

Pg.431

Find the values of x and y for the quadrilateral. Justify your

answer.

For a rectangle, diagonals must bisect and be congruent; and opposite sides are congruent.

Therefore, using the SSS theorem, the triangles are

congruent and CPCTC says…

x = 25

The corners of a rectangle are right angles.

Therefore…x + y = 90

25 + y = 90y = 65

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Exercises:Exercises:

#15

Pg.431

Find the value of x for the quadrilateral. Justify your

answer.

For a square, the diagonals will bisect each angle and each

corner will = 90o.

Therefore, 2x – 6 + 2x – 6 = 90

4x = 90 + 12

4x = 102

x = 25.5

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Homework: Practice 6-2CQuiz Friday