Quadrilateral FamilyTopic Index | Geometry Index | Regents Exam Prep Center

Each member of the quadrilateral family will describe its specific properties.

*QuadrilateralI have exactly four sides. The sum of the interior angles of all quadrilaterals is 360.

A quadrilateral is any four sided figure. Do not assume any additional properties for a quadrilateral unless you are given additional information.

*TrapezoidI have only one set of parallel sides. [The median of a trapezoid is parallel to the bases and equal to one-half the sum of the bases.]

A trapezoid has ONLY ONE set of parallel sides. When proving a figure is a trapezoid, it is necessary to prove that two sides are parallel and two sides are not parallel.

*Isosceles

Trapezoid

I have: - only one set of parallel sides - base angles congruent - legs congruent - diagonals congruent - opposite angles supplementary

Never assume that a trapezoid is isosceles unless you are given (or can prove) that information.

*ParallelogramI have: - 2 sets of parallel sides - 2 sets of congruent sides - opposite angles congruent - consecutive angles supplementary - diagonals bisect each other - diagonals form 2 congruent triangles

Notice how the properties of a parallelogram come in sets of twos: two properties about the sides; two properties about the angles; two properties about the diagonals. Use this fact to help you remember the properties.

*RectangleI have all of the properties of the parallelogram PLUS - 4 right angles - diagonals congruent

If you know the properties of a parallelogram, you only need to add 2 additional properties to describe a rectangle.

*RhombusI have all of the properties of the parallelogram PLUS - 4 congruent sides - diagonals bisect angles - diagonals perpendicular

A rhombus is a slanted square. It has all of the properties of a parallelogram plus three additional properties.

The square is the most specific Hey, look at me! member of the quadrilateral family. I have all of the properties of the It has the largest parallelogram AND the rectangle AND number of the rhombus. properties. I have it all!

*Square

Topic Index | Geometry Index | Regents Exam Prep Center Created by Donna Roberts Copyright 1998-2010 http://regentsprep.org Oswego City School District Regents Exam Prep Center

Theorems Dealing with ParallelogramsTopic Index | Geometry Index | Regents Exam Prep Center

*ParallelogramI have: - 2 sets of parallel sides - 2 sets of congruent sides - opposite angles congruent - consecutive angles supplementary - diagonals bisect each other - diagonals form 2 congruent triangles

Definition: A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

Using this definition, the remaining properties regarding a parallelogram can be "proven" true and become theorems.

When GIVEN a parallelogram, the definition and theorems are stated as ...A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

If a quadrilateral is a parallelogram, the 2 pairs of opposite sides are congruent.(Proof appears further down the page.)

If a quadrilateral is a parallelogram, the 2 pairs of opposite angles are congruent. If a quadrilateral is a parallelogram, the consecutive angles are supplementary. If a quadrilateral is a parallelogram, the diagonals bisect each other. If a quadrilateral is a parallelogram, the diagonals form two congruent triangles.

When trying to PROVE a parallelogram, the definition and theorems are stated as ...(many of these theorems are converses of the previous theorems)

A parallelogram is a quadrilateral with both pairs of opposite sides parallel. If both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram. If both pairs of opposite angles of a quadrilateral are congruent, the quadrilateral is a parallelogram. If one angle is supplementary to both consecutive angles in a quadrilateral, the quadrilateral is a parallelogram. If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram.

If ONE PAIR of opposite sides of a quadrilateral are BOTH parallel and congruent, the quadrilateral is a parallelogram. (Proof appearsfurther down the page.)

** Be sure to remember this last method, as it may save you time whenworking a proof.

Proof of Theorem:

If a quadrilateral is a parallelogram, the 2 pairs of opposite sides are congruent.

(Remember: when attempting to prove a theorem to be true, you cannot use the theorem as a reason in your proof.)

STATEMENTS

REASONS

1 2 Draw segment from A to C 3 4

1 Given 2 Two points determine exactly one line. 3 A parallelogram is a quadrilateral with both pairs of opposite sides parallel. 4 If two parallel lines are cut by a transversal, the alternate interior angles are congruent. 5 Reflexive property: A quantity is congruent to itself. 6 ASA: If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. 7 CPCTC: Corresponding parts of congruent

5 6

7

triangles are congruent.

Proof of Theorem:

If ONE PAIR of opposite sides of a quadrilateral are BOTH parallel and congruent, the quadrilateral is a parallelogram.

(Remember: when attempting to prove a theorem to be true, you cannot use the theorem as a reason in your proof.)

STATEMENTS

REASONS

1 2 Draw segment from A to C 3 4 5

1 Given 2 Two points determine exactly one line. 3 If two parallel lines are cut by a transversal, the alternate interior angles are congruent. 4 Reflexive property: A quantity is congruent to itself. 5 SAS: If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. 6 CPCTC: Corresponding parts of congruent triangles are congruent. 7 If two lines are cut by a transversal and the alternate interior angles are congruent, the lines are parallel. 8 A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

6 7

8

Theorems Dealing with Rectangles, Rhombuses, SquaresTopic Index | Geometry Index | Regents Exam Prep Center

Definition: A rectangle is a parallelogram with four right angles.

Using the definition, the properties of the rectangle can be "proven" true and become theorems.

When dealing with a rectangle, the definition and theorems are stated as ... 1.A rectangle is a parallelogram with four right angles.

*RectangleI have all of the properties of the parallelogram PLUS - 4 right angles - diagonals congruent

While the definition states "parallelogram", it is sufficient to say: "A quadrilateral is a rectangle if and only if it has four right angles.", since any quadrilateral with four right angles is a parallelogram.

2. 3.

If a parallelogram has one right angle it is a rectangle.

A parallelogram is a rectangle if and only if its diagonals are congruent.

Construction workers use the fact that the diagonals of a rectangle are congruent (equal) when attempting to build a "square" footing for a building, a patio, a fenced area, a table top, etc. Workers measure the diagonals. When the diagonals of the project are equal the building line is said to be square.

Definition: A rhombus is a parallelogram with four congruent sides.

Using the definition, the properties of the rhombus can be "proven" true and become theorems.

When dealing with a rhombus, the definition and theorems are stated as ... 1. A rhombus is a parallelogram with fourcongruent sides.While the definition states "parallelogram", it is sufficient to say: "A quadrilateral is a rhombus if and only if it has four congruent sides.", since any quadrilateral with four congruent sides is a parallelogram.

*RhombusI have all of the properties of the parallelogram PLUS - 4 congruent sides - diagonals bisect angles - diagonals perpendicular

2. If a parallelogram has two consecutive sidescongruent, it is a rhombus.

3. A parallelogram is a rhombus if and only if eachdiagonal bisects a pair of opposite angles.

4. A parallelogram is a rhombus if and only if thediagonals are perpendicular.(Proof of theorem appears further down page.)

Definition: A square is a parallelogram with four congruent sides and four right angles.

Using the definition, the properties of the rhombus can be "proven" true and become theorems.

When dealing with a square, the definition is stated as ...

A square is a parallelogram with four congruent sides and four right angles.This definition may also be stated as

A quadrilateral is a square if and only if it is a rhombus and a rectangle.

*SquareHey, look at me! I have all of the properties of the parallelogram AND the rectangle AND the rhombus. I have it all!

Proof of Theorem:

If a parallelogram is a rhombus, then the diagonals are perpendicular.

(Remember: when attempting to prove a theorem to be true, you cannot use the theorem as a reason in your proof.)

STATEMENTS

REASONS

1 2 Draw segment from A to C 3

1 Given 2 Two points determine exactly one line. 3 A rhombus is a parallelogram with four congruent sides.

4 5 6 7

4 A rhombus is a parallelogram with four congruent sides. 5 If a quadrilateral is a parallelogram, the diagonals bisect each other. 6 A bisector of a segment intersects the segment at its midpoint. 7 Midpoint of a line segment is the point on that line segment that divides the segment two congruent segments. 8 Reflexive Property - A quantity is congruent to itself. 9 SSS - If three sides of one triangle are congruent to three sides of a second triangle, the triangles are congruent. 10 CPCTC - Corresponding parts of congruent triangles are congruent. 11 If 2 congruent angles form a linear pair, they are right angles. 12 Perpendicular lines meet to form right angles.

8 9

10 11 12

Topic Index | Geometry Index | Regents Exam Prep Center Created by Donna Roberts Copyright 1998-2010 http://regentsprep.org Oswego City School District Regents Exam Prep Center

Theorems Dealing with TrapezoidsTopic Index | Geometry Index | Regents Exam Prep Center

Definition: A trapezoid is a quadrilateral with exactly one pair of parallel sides.

*TrapezoidI have only one set of parallel sides. [The median of a trapezoid is parallel to the bases and equal to one-half the sum of the bases.]

A trapezoid has ONLY ONE set of parallel sides. When proving a figure is a trapezoid, it is necessary to prove that two sides are parallel and two sides are NOT parallel.

The median (also called the mid-segment) of a trapezoid is a segment that connects the midpoint of one leg to the midpoint of the other leg. Theorem: The median (or mid-segment) of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases. (True for ALL trapezoids.)

;

Definition: An isosceles trapezoid is a trapezoid with congruent legs.

Theorems: 1. A trapezoid is isosceles if and only if the baseangles are congruent.

2. A trapezoid is isosceles if and only if the

*Isosceles

Trapezoid

diagonals are congruent.

I have: - only one set of parallel sides - base angles congruent - legs congruent - diagonals congruent - opposite angles supplementary

3.

If a trapezoid is isosceles, the opposite angles are supplementary.Never assume that a trapezoid is isosceles unless you are given (or can prove) that information.

Topic Index | Geometry Index | Regents Exam Prep Center Created by Donna Roberts Copyright 1998-2010 http://regentsprep.org Oswego City School District Regents Exam Prep Center