Quadrilaterals

Chapter 5

Pre-AP Geometry

Objectives

Apply the definition of a parallelogram and the theorems about properties of a parallelogram.

Prove that certain quadrilaterals are parallelograms. Apply theorems about parallel lines. Apply the midpoint theorems for triangles. Apply the definitions and identify the special properties of

a rectangle, a rhombus, and a square. Determine when a parallelogram is a rectangle,

rhombus, or square. Apply the definition and identify the properties of a

trapezoid and an isosceles trapezoid.

Parallelograms

Lesson 5.1

Pre-AP Geometry

Objective

Apply the definition of a parallelogram and the theorems about properties of a parallelogram.

Definition

Parallelogram A quadrilateral with two sets of parallel sides.

Properties of Parallelograms

The diagonals of a parallelogram bisect each other.

Opposite sides of a parallelogram are congruent.

Opposite angles of a parallelogram are congruent.

Each diagonal bisects the parallelogram into two congruent triangles.

Parallelogram

Theorems

5-1 Opposite sides of a parallelogram are congruent.

5-2 Opposite angles of a parallelogram are congruent.

5-3 Diagonals of a parallelogram bisect each other.

Practice1. Name all pairs of parallel lines.2. Name all pairs of congruent angles.3. Name all pairs of congruent segments.

4. What is the sum of the measures of the interior angles of a parallelogram?

5. What is the sum of the measures of the exterior angles of a parallelogram?

Review – True or False

1. Every parallelogram is a quadrilateral.

2. Every quadrilateral is a parallelogram.

3. All angles of a parallelogram are congruent.

4. All sides of a parallelogram are congruent.

Written Exercises

Problem Set 5.1, p. 169: # 2 – 32 (even) skip # 16

Proving Quadrilaterals are Parallelograms

Lesson 5.2

Pre-AP Geometry

Objective

Prove that certain quadrilaterals are parallelograms.

Quadrilaterals and Parallelograms

A quadrilateral is a polygon with 4 sides. A parallelogram is a quadrilateral whose opposite

sides are parallel (the top and bottom are parallel and the left and right are parallel).

Theorem 5-4

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

Theorem 5-5

If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram.

Theorem 5-6

If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Theorem 5-7

If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Five ways to Prove that a Quadrilateral is a Parallelogram

1. Show that both pairs of opposite sides are parallel.

2. Show that both pairs of opposite sides are congruent.

3. Show that one pair of opposite sides are both congruent and parallel.

4. Show that both pairs of opposite angles are congruent.

5. Show that the diagonals bisect each other.

Review

Answer with always, sometimes, or never.

1. The diagonals of a quadrilateral bisect each other.

2. If the measures of two angles of a quadrilateral are equal, then he quadrilateral is a parallelogram.

3. If one pair of opposite sides of a quadrilateral is congruent and parallel, then the quadrilateral is a parallelogram.

4. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

5. To prove a quadrilateral is a parallelogram, it is enough to show that one pair of opposite sides is parallel.

Practice

State the definition of theorem that enables you to deduce, from the information provided, that quadrilateral ABCD is a parallelogram.

1. BE = EX; CE = EA

2. BAD DCB; ADC CBA

3. BC || AD; AB || DC

4. BC AD; AB DCA

B C

D

E

Written ExercisesProblem Set 5.2 p. 174 # 2, 4, 8, 10, 14, 20,

22

Theorems Involving Parallel Lines

Lesson 5.3

Pre-AP Geometry

Objective

Apply theorems about parallel lines and the segment that joins the midpoints of two sides of a triangle.

Theorem 5-8

If two lines are parallel, then all points on one line are equidistant from the other line.

l

m

Theorem 5-9

If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.

l

m

n

Theorem 5-10

A line that contains the midpoint of one side of a triangle and is parallel to another side passes through the midpoint of the third side.

D

B

NM

A

C

Theorem 5-11

The segment that joins the midpoints of two sides of a triangle:(1) is parallel to the third side;(2) is half as long as the third side.

B

NM

A

C

BM = AM, CN = ANBC = 2(MN)MN = ½(BC)

Written Exercises

Problem Set 5.3 p. 180 # 2-20 even, p. 182 # 1-6

Special Parallelograms

Lesson 5.4

Pre-AP Geometry

Objectives

Apply the definitions and identify the special properties of a rectangle, a rhombus, and a square.

Determine when a parallelogram is a rectangle, rhombus, or square.

A parallelogram with four right angles.

Both pairs of opposite angles are congruent.

Every rectangle is a parallelogram.

Rectangle

A quadrilateral with four congruent sides.

Both pairs of opposite sides are congruent.

Every rhombus is a parallelogram.

Rhombus

A quadrilateral with four right angles and four congruent sides.

Both pairs of opposite angles and opposite sides are congruent.

A square is also a rectangle, a rhombus, and a parallelogram.

Square

Theorem 5-12

The diagonals of a rectangle are congruent.

Theorem 5-13

The diagonals of a rhombus are perpendicular.

Theorem 5-14

Each diagonal of a rhombus bisects two angles of the rhombus.

Theorem 5-15

The midpoint of the hypotenuse of a right angle is equidistant from the three vertices.

Theorem 5-16

If an angle of a parallelogram is a right angle, then the parallelogram is a rectangle.

Theorem 5-17

If two consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus.

Practice

Reply with always, sometimes, or never.

1. A square it a rhombus.2. The diagonals of a parallelogram bisect the angles of the

parallelogram.3. A quadrilateral with one pair of sides congruent is a

parallelogram.4. The diagonals of a rhombus are congruent.5. A rectangle has consecutive sides congruent.6. A rectangle has perpendicular diagonals.7. The diagonals of a rhombus bisect each other.8. The diagonals of a parallelogram are perpendicular bisectors of

each other.

Written Exercises

Problem Set 5.4 p. 187 # 1-10, 12-30 evens

Trapezoids

Lesson 5.5

Pre-AP Geometry

Objective

Apply the definitions and identify the properties of a trapezoid and an isosceles trapezoid.

Definition

Trapezoid

A quadrilateral with exactly one pair of parallel sides.

leg leg

base

base

DefinitionIsosceles Trapezoid

In an isosceles trapezoid, the base angles are equal, and so are the other pair of opposite sides AD and BC.

A B

CD

Theorem 5-18

Base angles of an isosceles trapezoid are congruent.

Median of a Trapezoid

The segment that joins the midpoints of the legs of a trapezoid.

Median of a Trapezoid

Theorem 5-19

The median of a trapezoid:

(1) is parallel to the base;

(2) has a length equal to the average of the base lengths.

Practice

In trapezoid ABCD, EF is a median.

1. If AB = 25 and DC = 13, then EF = _____.2. If AE = 11 and FB = 8, then AD = _____ and BC =

_____.3. If AB = 29 and EF = 24, then DC = _____.4. If AB = 7y + 6 and EF = 5y – 3 and DC = y – 5, then y = _____.

Written Exercises

Problem Set 5.5 p. 192 # 2-26 evens