SECTION 5.5

Properties of Quadrilaterals

Opposite sides are parallel ( DC ll AB, AD ll BC )

Opposite sides are congruent ( DA CB, DC AB )

Opposite angles are congruent (<DAB <DCB, <ABC <ADC)

Diagonals bisect each other (DB bis. AC, AC bis. DB)

Consecutive angles are supplementary(<DAB suppl. <ADC, etc.)

Diagonals form 2 congruent triangles ( ABC CDA, DCB BAD)

Properties of Parallelograms

PROPERTIES OF A RECTANGLE All properties of a parallelogram apply

All angles are right angles and .

Diagonals are ( )

AB

C D

CABD

PROPERTIES OF A KITE Two disjoint pairs of

consecutive sides are

Diagonals are One diagonal is the

bisector of the other One of the diagonals

bisect a pair of opposite <‘s

One pair of opposite <‘s are

A

B

C

D

PROPERTIES OF A RHOMBUS

Parallelogram Properties

Kite Properties All sides are

congruent Diagonals bisect the

angles Diagonals are

perpendicular bisectors of each other

Diagonals divide the rhombus into 4 congruent rt. Triangles

PROPERTIES OF SQUARES Rectangle Properties Rhombus Properties Diagonals form 4 isos. right triangles

PROPERTIES OF TRAPEZOIDS

Exactly one pair of sides parallel

PROPERTIES OF ISOSCELES TRAPEZOIDS

Legs are congruent Bases are parallel Lower base angles are congruent Upper base angles are congruent Diagonals are congruent Lower base angles are suppl. to upper

base angles

Always, sometimes, neverThe diagonals of a rectangle are congruent Every square is a rectangle Every quadrilateral is a trapezoidIn a trap. opp angles are congruentA rhombus is a rectangleAn isos. trap is parallelogramConsecutive angles of a square are congruentRhombuses are parallelogramsSquares have only one right angleNo trapezoid is a rectangleAn isosceles trapezoid has no parallel lines

AlwaysAlways

Sometimes

Never

Sometimes

Never

Always

Always

Never

Always

Never

Practice Problems

SAMPLE PROBLEMS

Statements Reasons

Given: Triangle ACE is isos. With base AE CD CB AG FE BD GFProve: BGFD is a parallelogram

A

B

C

D

EFG

1. tri. ACE is isos. w/ base AE

2. CD CB3. AG FE4. BD GF

1. Given2. Given3. Given4. Given

5. <A <E 5. If isos, then <‘s6. CA CE 6. If <‘s, then sides7. BA DE 7. Subtraction

8. Tri. BAG Tri. DEF 8. SAS(3,5,7)9. BG DF 9. CPCTC

10. BGDF is a parallelogram 10. If opp. sides are then figureis a parallelogram

SAMPLE PROBLEM #2

E

Given: ABCD is a rhombus

Prove: AC is perp. DB

SAMPLE PROBLEM #2

Statement Reason

1. ABCD is a rhombus2. AD DC3. DE DE4. AE CE5. Tri. ADE and Tri. CDE6. <AED <CED7. <AED and <CED are rt<s8. AC DB

E

1. Given2. In a rhombus opp. Sides are 3. Reflexive4. In a parallelogram diag. bisect each other5. SSS(2,3,4)6. CPCTC7. If 2 <s are and suppl. They are rt. <s.8. Rt <s are formed by perp. lines

Characteristics parallelogram

rhombus rectangle square trapezoid Isosceles trapezoid

kite

Both pairs of opp sides ll

1/2

Diag.

Both pairs of opp sides are

1/2

At least 1 rt <

Both pairs of opp. <s

Cons. <s suppl

1/2

Diag form 2 tri.

1/2Exactly 1 pair of opp. sides ll

Diag. perp.

Consecutive sides

1/2

Consecutive <s

Diagonals bisect e.o.

1/2

Diagonals bisect opp. <s

1/2

All sides

All <‘s

"Quickie Math." Quickie Math , n.d. Web. 19 Jan 2011. <library.thinkquest.org/C006354/11_1.html>.

“Rhombus problems." analyze math. A Dendane , 5 November 2010. Web. 19 Jan 2011.

<http://www.analyzemath.com/Geometry/rhombus_problems.html>.

Works Cited

Rhoad, Richard, George Miluaskas, and Robert Whipple. Geometry for Enjoyment and Challenge. New Edition ed. Boston:

McDougal Littell, 1997. Print.