Geometry CC Review – Quadrilaterals
Properties of quadrilaterals
- Quadrilateral
o 4 sides, sum of interior angles, sum of exterior angles
- Parallelogram
o Opposite sides parallel, opposite sides  , opposite angles  , diagonals bisect each other, consecutive angle
supplementary
- Rectangle
o Parallelogram PLUS contains 4 right angles,  diagonals
- Rhombus
o Parallelogram PLUS 4  sides, perpendicular diagonals, diagonals bisect angles
- Square
o Parallelogram PLUS rhombus and a rectangle
*** Rhombus and a square both have perpendicular diagonals (forms 4 right triangles and allows for the use of Pythagorean
Theorem).
-
-
Trapezoid
o one pair of opposite sides are parallel (bases) and one pair of opposite sides are not parallel
o median of trapezoid is parallel to the bases and equals one-half the sum of the bases
Isosceles trapezoid
o non parallel sides are congruent, base angles are congruent, diagonals are congruent, opposite angles are
supplementary
Quadrilateral inscribed in a circle – opposite angles are supplementary.
Proofs
distance: d  ( x2  x1 ) 2  ( y 2  y1 ) 2
 x1  x2 y1  y 2 
,

2 
 2
midpoint: M  
Prove a quadrilateral is a parallelogram by showing:
o both pairs of opposite sides are parallel (slope) OR
o both pairs of opposite sides are congruent (distance) OR
o one pair of opposite sides is both parallel and congruent (slope and distance) OR
o the diagonals bisect each other (midpoint)
Prove a quadrilateral is a rectangle by showing:
o a parallelogram with one right angle (slopes are negative reciprocals)
o a parallelogram with congruent diagonals (distance)
slope: m 
June 2015
#13, 26 33 and 36
August 2015
#1, 15, 22, 28, and 35
January 2016
#3 and 35
Prove a quadrilateral is a rhombus by showing:
o 4 congruent sides (distance)
o a parallelogram with perpendicular diagonals (slopes are negative reciprocals)
o a parallelogram with 2 congruent adjacent sides (distance)
Prove a quadrilateral is a square by showing:
o a rectangle with 2 congruent adjacent sides (distance)
o a rhombus with one right angle (slopes are negative reciprocals)
Prove a quadrilateral is a trapezoid by showing:
o one pair of opposite sides are parallel AND one pair of opposite sides are not parallel (slope)
Prove a quadrilateral is an isosceles trapezoid by showing
o a trapezoid with congruent leg (distance)
o a trapezoid with congruent diagonals (distance)
y y 2  y1

x x 2  x1
June 2015 #33
Statements
1. Parallelogram ABCD
Reason
1. Given
2. AD  CB
2. Opposite sides of a parallelogram are congruent
(side)
3. AD || CB
4. ADE  CBE (angle)
5. AED  CEB (angle)
6. AED  CEB
3. Opposite sides of a parallelogram are parallel
4. If 2 parallel lines are cut by a transversal, the alternate interior angles are
congruent
5. Opposite angles formed by intersecting lines are vertical angles and all
vertical angles are congruent
6. AAS  AAS
August 2015 #35
Statements
1. Parallelogram ABCD
BE  CED , DF  BFC
2. DFC  BEC (angle)
Reason
1. Given
3. CE  CF (side)
4. C  C (angle)
5. BEC  DFC
3. Given
4. Reflexive property
5. ASA  ASA
6. BC  DC
7. ABCD is a rhombus
6. CPCTC
7. If the consecutive sides of a parallelogram are congruent, it’s a rhombus
2. Perpendicular lines form right angles and all right angles are congruent.
January 2016 #35
Statements
1. Parallelogram ANDR
Reason
1. Given
2. AW bisects NWD
DE bisects REA
3. AE  RE
2. Given
4.
5.
6.
7.
NW  DW
RA  DN
RD  NA (side)
NW  RE (side)
R  N (angle)
RED  NWA
ED  WA
AE  DW
8.
9.
10. Quadrilateral AWDE is a parallelogram
3. Segment bisector divides a line segment into two congruent segments
4. Opposite sides of a parallelogram are congruent
5. Halves of congruent segments are congruent
6. Opposite angles of a parallelogram are congruent
7. SAS  SAS
8. CPCTC
9. Halves of congruent segments are congruent
10. If both pairs of opposite sides are congruent, the
quadrilateral is a parallelogram