### Geometry Chapter 6 review

 Parallelograms  Rectangles  Squares and Rhombi  Trapezoids

### Parallelograms

• Definition - A quadrilateral with two pairs of parallel sides • Opposite sides are congruent • Opposite angles are congruent • Consecutive angles are supplementary

### Parallelograms

• The diagonals bisect each other

AX

@

CX

,

BX

@

DX

A B X D C

### A quadrilateral is a parallelogram if:

• Both pairs of opposite sides are parallel • If both pairs of of opposite sides are congruent • If both pairs of opposite angles are congruent • If the diagonals bisect each other • If one pair of opposite sides are congruent and parallel • All consecutive angles are supplementary

Find the values of x & y that make the quadrilateral a parallelogram 64 5y 2y + 36 6x – 2 5y = 2y + 36 3y = 36 Y = 12 6x – 2 = 64 6x = 66 x = 11

Find the values of x & y that make the quadrilateral a parallelogram 4x 12y 96 76 12y = 96 Y = 8 4x = 76 x = 19

Find the values of x & y that make the quadrilateral a parallelogram 3x + 17 4 4x – y 3x + 17 = 2y 2y 4x – y = 4 4x – 4 = y 3x + 17 = 2(4x – 4) 3x + 17 = 8x – 8 -5x = -25 4(5) – 4 = y x = 5 y = 16

Find the values of x & y that make the quadrilateral a parallelogram 2y 4x 7x y + 2 4x = y + 2 4x – 2 = y 2y = 7x 2(4x – 2) = 7x 8x – 4 = 7x -4 = -x x = 4 4(4) - 2 = y 14 = y

Determine if whether quadrilateral ABCD is a parallelogram A (5,6) B(9,0), C(8,-5), D (3,-2) A (5,6) B(9,0) D(3,-2) C(8,-5) By slope: AB = (0 - 6)/(9 - 5) = -6/4 DC = (-5 - -2)/(8 - 3) = -3/5 Not parallel, therefore not a parallelogram

Determine if whether quadrilateral ABCD is a parallelogram A (-7,3) B(-3,2), C(0,-4), D (-4,-3) A (-7,3) B(-3/2) D(-4,-3) C(0,-4)

AB

=

DC

=

=

BC

= ( 3 - 7) 2 + (2 3) 2 = ( 4 - 3) 2 + (0 - 4) 2 = 4 2 + 1 2 = 1 2 + 4 2 = 17 17 (3 - 3) 2 + ( 4 - 7) 2 = ( 3 0) 2 + (2 - 4) 2 = 6 2 + 3 2 3 2 + 6 2 = = 45 45

### Rectangles

• Definition - A quadrilateral with four right angles • The diagonals of a rectangle are congruent • If the diagonals of a parallelogram are congruent, then it is a rectangle.

A B X D C

### Rectangles

A B X D C

AC

@

DB

Diagonals of a rectangle are @

AX

@

XC

Diagonals of a parallelogram bisect each other.

DX

@

XB

,

AX

@

XC

Diagonals of a parallelogram bisect each other.

AX

@

DX

and other sides of all triangles Proved using Segment Additive Postulate and substitution.

Angle

@ Angle

XDA

and all pairs of angles opposite congruent sides.

I soseles Triangle Theorem

### Rhombi

• A rhombus is a quadrilateral with 4 congruent sides.

• The diagonals of a rhombus are perpendicular, and • If the diagonals of a parallelogram are perpendicular, then you have a rhombus • The diagonals bisect a pair of opposite angles.

A B D C

A B

### Given Rhombi ABCD

F D C If angle BAF = 28, find angle ACD Alternate Interior angles 28 Find x if angle AFB = 16x + 10 16x + 10 = 90 x = 5 If angle ACD = 34, find angle ABC 180 – 34 – 34 = 112 What is the value of x if BAC = 4x + 6 and ACD = 12x – 18 4x + 6 = 12x – 18 x = 3 If DCB = x 2 – 6 and DAC = 5x + 9, find x.

x 2 – 6 = 2(5x + 9), find x. x 2 – 10x – 24 = 0 (x – 12)(x + 2) =0; 12

### Squares

• A square is a quadrilateral with 4 right angles and 4 congruent side.

• A square is both a rectangle and a rhombus. All properties of rectangles and rhombi apply to squares.

45˚ 45˚ Diagonals are congruent, perpendicular and bisected.

All right angles are bisected.

Property Diagonals bisect each other The diagonals are congruent Parallelogram Yes No Each diagonal bisect a pair of opposite angles.

The diagonals are perpendicular No No Rectangle Yes Yes No No Rhombus Yes No Yes Square Yes Yes Yes Yes Yes

L 3 I 1 2 4 6 5 K J

### Homework

Given Rhombus IJKL, determine the following.

1) Angle 3 = 62, find angle 1 2) Angle 4 = 3x – 1, angle 3 = 2x + 30, find x.

3) Angle 5 = 2(x + 1), angle 3 = 4(x + 1), find x.

4) Angle 6 = 7x + 13, find x.

5) Angle LKJ = x 2 – 17, angle 2 = x + 23, find x.

I L M K J

### Homework

Given Square IJKL, determine the following.

6) IM = 4x + 12, JL = 9x – 4, find MK 7) If ML = 2, find IL 8) If IJ = 8, find angle MIJ

Determine whether quadrilateral WXYZ is a parallelogram, a rectangle, a rhombus, or a square for the following set of points. Name all that apply.

9) W(5,6), X(7,5), Y(9,9), Z(7,10) 10) W(10,6), X(6,10), Y(10,14), Z(14,10)

### Trapezoids

• A trapezoid is a quadrilateral with exactly one pair of parallel side. The parallel sides are called the bases. The non-parallel sides are called legs.

• An isosceles trapezoids has legs that are congruent.

base base leg leg leg leg base base

### Isosceles Trapezoids

• Both pairs of base angles of an isosceles triangle are congruent.

• The diagonals of an isosceles trapezoid are congruent.

### Isosceles Trapezoids

• The median of a trapezoid is the segment that connects the midpoints of the legs.

• The median is parallel to the bases, and its measure is equal to one-half the sum of the measures of the bases.

3x – 1 10 7x + 1 10 = 1/2(3x – 1 + 7x + 1) 10 = 1/2(10x) 10 = 5x x = 2

Given isosceles trapezoid PQRS with bases PS & QR and median TV. Find the following If PS = 20 and QR = 14, find TV 17 If QR = 14.3 and TV = 23.2, find PS 32.1

If TV = x + 7 and PS + QR = 5x + 2, find x 4 If angle RVT = 57˚, find angle QTV 57 If angle VTP = 112˚, find angle TPS 68˚ P S V T Q R