Advanced Geometry

First you must prove or be given that the figure is a parallelogram, then

A PARALLELOGRAM is a rectangle if…1. It contains at least one right angle,

2. Or if the diagonals are congruent

A QUADRILATERAL is a rectangle if…All FOUR angles are right angles

1 4

32

Two isn’t enough!

In this case, you DO NOT have to

prove parallelogram first. Notice that we are going from QUAD to RECTANGLE

here!

A QUADRILATERAL is a kite if…1. It has two pairs of consecutive

congruent sides, or2. If one of the diagonals is the

perpendicular

bisector of the other

A PARALLELOGRAM is a rhombus if…1. Parallelogram with a pair of consecutive

congruent sides ~ or ~2. Parallelogram where either diagonal bisects

two angles

A QUADRILATERAL is a rhombus if…

The diagonals are perpendicular bisectors of each other.

of each other

A QUADRILATERAL is a square if…It is both a rectangle and a rhombus

Both are Parallelograms, so: Opposite sides parallelOpposite angles congruentConsecutive angles supplementary

From Rhombus: ALL sides congruentDiagonals are perpendicular bisectors of each otherDiagonals bisect opposite angles

From Rectangle: 4 Right AnglesDiagonals are congruent

A QUADRILATERAL is a square if…It is both a rectangle and a rhombus

Both are Parallelograms, so: Opposite sides parallelOpposite angles congruentConsecutive angles supplementary

From Rhombus: ALL sides congruentDiagonals are perpendicular bisectors of each otherDiagonals bisect opposite angles

From Rectangle: 4 Right AnglesDiagonals are congruent

Each diagonal creates two 45⁰-45⁰-90⁰ISOSCELES, RIGHT TRIANGLES!

A TRAPEZOID is isosceles if… The nonparallel sides are congruent, or The lower or upper base angles are

congruent, 0r The diagonals are congruent

XX √√Trapezoid : X X Trapezoid : X X XX

Isosceles Trapezoid: Isosceles Trapezoid: √√√√√√

√√√√√√

Sample Problem 1: What is the most descriptive name for quadrilateral ABCD with vertices A(-3, -7) B(-9, 1) C(3, 9) D(9, 1)

Step 1: Graph ABCD Step 2: Find the slope of each side.

A(-3, -7)

(-9, 1)B

C(3, 9)

(9, 1) D

9 - 1

3 – (-9)

9 - 1

1- (-7)

-9- (-3)

3 - 9

-7- (1)

-3 – 9

Slope:Slope:

AB AB -8/6 = -4/3 -8/6 = -4/3BC BC 8/12 = 2/3 8/12 = 2/3CD CD 8/-6 = -4/3 8/-6 = -4/3AD AD -8/-12 = 2/3 -8/-12 = 2/3

8

12 - 6

8

- 8

- 8

-12 6

Sample Problem 1: What is the most descriptive name for quadrilateral ABCD with vertices A(-3, -7) B(-9, 1) C(3, 9) D(9, 1)

Step 3: •compare the slopes of opposite sides to determine whether they are parallel.

Slope AB = Slope CD (- 4/3) and Slope BC = Slope AD (2/3)

SAME SLOPES Opposite sides PARALLEL Parallelogram!

• compare the slopes of consecutive sides to see if they are perpendicular. (slope AB)(slope BC) = (-4/3)(2/3) ≠ -1,

so consecutive sides are not

OPPOSITE RECIPROCAL SLOPES Consecutive sides PERPENDICULAR Rectangle

Step 4: We know the quadrilateral is not a rectangle, but it is a parallelogram. Find the slopes of the diagonals to see if it is a rhombus.

Sample Problem 1: What is the most descriptive name for quadrilateral ABCD with vertices A(-3, -7) B(-9, 1) C(3, 9) D(9, 1)

Step 4: We now know the quadrilateral is not a rectangle, but IT IS A PARALLELOGRAM. Find the slopes of the diagonals to see if it is a rhombus.

(-9, 1)B

C(3, 9)

A(-3, -7)

(9, 1) D

Slope BD = (1 – 1)/[9 – (-9)] = 0/18 = 0

Slope AC = [9 – (-7)] / [3 – (-3)] = 16/6 = 8/3

(Slope BD)(Slope AC) = (0)(8/3) ≠ -1

Since the slopes are not opposite reciprocals, the diagonals are not

ABCD is a PARALLELOGRAM

In a rhombus, the diagonals are ,

Given: AB || CD ∡ABC ∡ADC AB AD

PROVE: ABCD is a rhombus

A

CB

D|| lines alt int ∡s , so ∡ABD ∡CDB

AB = AD was not used yet, because

FIRST we must prove that ABCD is a parallelogram,

then we will try to prove that it is a

RHOMBUS!

Given: AB || CD ∡ABC ∡ADC AB AD

PROVE: ABCD is a rhombus

A

CB

D∡CBD ∡ADB Subtraction!

∡ABC - ∡ABD = ∡ CBD∡ADC - ∡CDB = ∡ADB

Given: AB || CD ∡ABC ∡ADC AB AD

PROVE: ABCD is a rhombus

A

CB

D

alt int ∡s || linesBC || AD

Given: AB || CD ∡ABC ∡ADC AB AD

PROVE: ABCD is a rhombus

A

CB

D

ABCD is a parallelogram

If a quadrilateral has 2 pairs of || sides, then parallelogram

Given: AB || CD ∡ABC ∡ADC AB AD

PROVE: ABCD is a rhombus

A

CB

D

ABCD is a rhombus

If a parallelogram has at least 2 consecutive sides , then rhombus!parallelogr

am

1. AB || CD

8. ABCD is a rhombus

A

CB

D

6. ABCD is a parallelogram

If a parallelogram has at least 2 consecutive sides , then RHOMBUS!

3. ∡ABC ∡ADC

7. AB AD

GIVEN

GIVEN2. ∡ABD ∡CDB

|| lines alt int ∡s

4. ∡ADB ∡CBD

subtraction5. BC || AD

alt int ∡s || lines If a quad has 2 pairs of || sides, then parallelogramGIVEN

Statements Reasons

parallelogram

then RHOMBUS !

Pp. 258 – 262

(1 – 6; 10; 12 – 14; 16, 17; 19 – 21; 24, 28, 29)