GEOMETRY R Unit 10 Quadrilaterals

MR. QUILL

Date Classwork Day Assignment

Monday 1/9

Parallelograms – 10.1

1 Worksheet 10.1

Tuesday 1/10

Proving a Parallelogram - 10.2

2 Worksheet 10.2

Wednesday 1/11

The Rectangle – 10.3

3 Worksheet 10.3

Thursday 1/12

The Rhombus – 10.4 UNIT 10 QUIZ 1

4 Worksheet 10.4

Friday 1/13

The Square – 10.5

5 Midterm Review #1

Monday 1/16

NO SCHOOL

1/17-1/23 MIDTERM REVIEW MIDTERM: 1/24 8AM

Worksheet 10.5 Due 1/30

Monday 1/30

The Trapezoid – 10.6 UNIT 10 QUIZ 2

6 Worksheet 10.6

Tuesday 1/31

Coordinate Geometry Proof 7 Worksheet 10.7

Wednesday 2/1

Coordinate Geometry Proof 8 Worksheet 10.8

Thursday 2/2

Review UNIT QUIZ 3

9 Review Packet

Friday 2/3

Review (1/2 Day) 10 Review Packet

Monday 2/6

Review 11 Review Packet

Tuesday 2/7

UNIT 10 TEST 12

Name: ______________________________________________ Date: _________________

PARALLELOGRAMS

A parallelogram is a quadrilateral with both pairs of opposite sides parallel. To name a parallelogram, use the symbol _______________. In parallelogram ABCD, and

by definition.

Properties of Parallelograms Theorems Example Figure

Opposite sides of a parallelogram are

congruent.

Opposite angles of a

parallelogram are congruent.

Consecutive Angles in a

parallelogram are supplementary.

If a parallelogram has one right angle, then it has four

right angles.

Example 1. In parallelogram ABCD, suppose m = 55, AB = 2.5 feet, and BC = 1 foot. Find each

measure.

a. DC

b. m

c. m

Diagonals of Parallelograms Theorem Example Figure

Diagonals of a parallelogram bisect each other

A diagonal separates a parallelogram into two

congruent triangles.

Examples 2. If QRST is a parallelogram, find the value of the indicated variable.

a. x

b. y

c. z 3. Determine the coordinates of the intersection of the diagonals of parallelogram FGHI with

vertices F(-2, 4), G(3, 5), H(2, -3), and J(-3, -4).

TESTS FOR PARALLELOGRAMS

If a quadrilateral has each pair of opposite sides parallel, it is a parallelogram by definition. This is not the only test, however, that can be used to determine if a quadrilateral is a

parallelogram.

Conditions for Parallelograms Theorem Example Figure

If both pairs of opposite

sides are congruent, then the quadrilateral is a

parallelogram.

If both pairs of opposite

angles are congruent, then the quadrilateral is a

parallelogram.

If diagonals bisect each

other, then the quadrilateral is a parallelogram.

If one pair of opposite sides

is congruent and parallel, then the quadrilateral is a

parallelogram.

Examples Determine whether each quadrilateral is a parallelogram. Justify your answer. 1. 2. 3.

Find x and y so that each of the following quadrilaterals are parallelograms. 4. FK = 3x – 1, KG = 4y + 3, JK = 6y – 2, and KH = 2x + 3 5. 6. Graph quadrilateral KLMN with vertices K(2, 3), L(8, 4), M(7, -2), and N(1, -3). Determine whether the quadrilateral is a parallelogram. Justify your answer using the Slope Formula.

RECTANGLES

A rectangle is a parallelogram with four right angles. By definition, a rectangle has the following properties:

1. All four angles are right angles 4. Consecutive angles are supplementary 2. Opposite sides are parallel and congruent 5. Diagonals bisect each other

3. Opposite angles are congruent

Diagonals of a Rectangle

Abbreviation Example Figure If a parallelogram is a rectangle, then its diagonals are congruent.

Examples

1. A rectangular park has two walking paths as shown. If PS = 180 meters and PR = 200 meters, find the following.

a. QT

b. If TS = 120 meters, find PR.

c. If = 64, find .

2. Quadrilateral JKLM is a rectangle. If = 2x + 4 and = 7x + 5, find x.

Proving Parallelograms are Rectangles Abbreviation Example Figure

If diagonals of a parallelogram

are congruent, then the parallelogram is a rectangle.

Examples

1. A community recreation center has created an outdoor dodge ball field. To be sure that it meets the ideal playing field requirements, they measure the sides of the field and its diagonals. If AB = 60 ft, BC = 30 ft, CD = 60 ft, AD = 30 ft, AC = 67 ft, and BD = 67 ft, explain how the recreation center can be sure that the playing field is rectangular.

2. Quadrilateral PQRS has vertices P(-5, 3), Q(1, -1), R(-1, -4), and S(-7, 0). Determine whether PQRS is a rectangle.

RHOMBI AND SQUARES A rhombus is a parallelogram with all four sides congruent. A rhombus has all the properties

of a parallelogram.

Diagonals of a Rhombus Theorem Example Figure

If a parallelogram is a

rhombus, then its diagonals are perpendicular

If a parallelogram is a

rhombus, then each diagonal bisects a pair of opposite

angles.

Examples The diagonals of rhombus FGHI intersect at K. Use the given information to find each measure or value.

d. If = 82, find .

e. If GH = x + 9 and JH = 5x – 2, find x.

A square is a parallelogram with four congruent sides and four right angles. Recall that a parallelogram with four right angles is a rectangle, and a parallelogram with four congruent sides is a rhombus. Therefore, a parallelogram that is both a rectangle and a rhombus is also

a square.

Conditions for Rhombi and Squares Theorem Example Figure

If the diagonals of a parallelogram are

perpendicular, then the parallelogram is a rhombus.

If one diagonal of a

parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus.

If one pair of consecutive

sides of a parallelogram are congruent, then the

parallelogram is a rhombus.

6.20 – If a quadrilateral is both a rectangle and a rhombus, then it is a square. Examples The key to the successful excavation of an archaelogical site is accurate mapping. How can archaeologists be sure that the region they have marked off is a 1-meter by 1-meter square? Determine whether parallelogram JKLM with vertices J (-7, -2), K (0, 4), L (9, 2), and M (2, -4) is a rhombus, a rectangle, or a square. List all that apply and explain.

TRAPEZOIDS A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are

called bases. The nonparallel sides are called legs. The base angles are formed by the base and one of the legs.

Isosceles Trapezoids

Theorem Example Figure

If a trapezoid is isosceles, then each pair of base angles

is congruent

If a trapezoid has one pair of congruent base angles, then it is an isosceles trapezoid.

A trapezoid is isosceles if

and only if its diagonals are congruent.

Examples 1. The speaker shown is an isosceles trapezoid. If = 85, FK = 8 inches, and JG = 19

inches, find each measure.

f.

g. KH

2. To save space at a square table, cafeteria trays often incorcporate trapezoids into their

design. If WXYZ is an isosceles trapezoid and = 45, WV = 15 cm, and VY = 10 cm, find each measure below. a. b.

b. c. XZ d. XV

The midsegment of a trapezoid is the segment that connects the midpoints of the legs of the

trapezoid. The theorem below relates the midsegment and the bases of a trapezoid.

Trapezoid Midsegment Theorem Theorem Example Figure

The midsegment of a

trapezoid is parallel to each base and its measure is one

half the sum of the lengths of the bases.

Examples 3. In the figure, is the midsegment of trapezoid FGJK. What is the value of x? 4. Trapezoid ABCD is shown below. If is parallel to , what is the x-coordinate of

point G? 5. Quadrilateral ABCD has vertices A(-3, 4), B(2, 5), C(3, 3), and D(-1, 0). Show that ABCD

is a trapezoid and determine whether it is an isosceles trapezoid.

CLASSIFYING QUADRILATERALS USING COORDINATE GEOMETRY

1. Determine the coordinates of the intersection of the diagonals of parallelogram FGHI with vertices F(-2, 4), G(3, 5), H(2, -3), and J(-3, -4).

2. Graph quadrilateral KLMN with vertices K(2, 3), L(8, 4), M(7, -2), and N(1, -3).

Determine whether the quadrilateral is a parallelogram. Justify your answer using the Slope Formula and Distance formula.

3. Quadrilateral PQRS has vertices P(-5, 3), Q(1, -1), R(-1, -4), and S(-7, 0). Determine whether PQRS is a rectangle.

4. Determine whether parallelogram JKLM with vertices J (-7, -2), K (0, 4), L (9, 2), and

M (2, -4) is a rhombus, a rectangle, or a square. List all that apply and explain.

X

Y

X

Y

COORDINATE GEOMETRY PROOFS (CONTINUED)

MIDPOINT: 1 2 1 2,2 2

x x y yMp

SLOPE: rise

run

y

x 2 1

2 1

y y

x x

DISTANCE: 2 2

2 1 2 1x x y y

A(-3, 4), B(2, 5), C(3,3), D(-1,0). Prove ABCD is a trapezoid. Is the trapezoid isosceles? F(-2, 4), G(3, 5), H(2, -3), J(-3, -4). Prove that quadrilateral FGHJ is a parallelogram.

X

Y

X

Y

A(0, 0), B(a, 0), C(b + a, c), D(b, c). Prove quadrilateral ABCD is a parallelogram. A(-1, -1), B(4, 0), C(5, 5), D(0, 4). Prove ABCD is a rhombus.

X

Y

X

Y

A(0, 0), B(a, 0), C(a, b), D(0, b). Prove that ABCD is a rectangle. A(0, 0), B(2a, 0), C(2b, 2c). Prove that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length.