Geometry Chapter 6 review

1
Geometry Chapter 6 review

• Parallelograms
• Rectangles
• Squares and Rhombi
• Trapezoids

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Parallelograms

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

3
Parallelograms

• The diagonals bisect each other

A
B
X
D
C
4
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

5
Find the values of x y that make the
quadrilateral a parallelogram
64
5y
2y 36
6x 2
5y 2y 36 6x 2 64 3y 36 6x 66 Y
12 x 11
6
Find the values of x y that make the
quadrilateral a parallelogram
4x
12y
96
76
12y 96 4x 76 Y 8 x 19
7
Find the values of x y that make the
quadrilateral a parallelogram
3x 17
4
4x y
2y
3x 17 2y 4x y 4 4x 4 y 3x 17
2(4x 4) 3x 17 8x 8 -5x -25 x 5 4(5)
4 y y 16
8
Find the values of x y that make the
quadrilateral a parallelogram
7x
4x
2y
y 2
4x y 2 2y 7x 4x 2 y 2(4x 2)
7x 8x 4 7x -4 -x x 4 4(4) - 2
y 14 y
9
Determine if whether quadrilateral ABCD is a
parallelogramA (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
10
Determine if whether quadrilateral ABCD is a
parallelogramA (-7,3) B(-3,2), C(0,-4), D (-4,-3)
A (-7,3)
B(-3/2)
D(-4,-3)
C(0,-4)
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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.

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Rectangles
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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
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Given Rhombi ABCD
A
B
F
C
D
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 x2 6 and DAC 5x 9,
find x. x2 6 2(5x 9), find x. x2 10x
24 0 (x 12)(x 2) 0 12
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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.

Diagonals are congruent, perpendicular and
bisected. All right angles are bisected.
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Homework
I
J
2
4
Given Rhombus IJKL, determine the following.
1
6
3
5
K
L

• Angle 3 62, find angle 1
• Angle 4 3x 1, angle 3 2x 30, find x.
• Angle 5 2(x 1), angle 3 4(x 1), find x.
• Angle 6 7x 13, find x.
• Angle LKJ x2 17, angle 2 x 23, find x.

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Homework
I
J
M
Given Square IJKL, determine the following.
K
L

• 6) IM 4x 12, JL 9x 4, find MK
• If ML 2, find IL
• If IJ 8, find angle MIJ

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Homework

• Determine whether quadrilateral WXYZ is a
  parallelogram, a rectangle, a rhombus, or a
  square for the following set of points. Name all
  that apply.
• W(5,6), X(7,5), Y(9,9), Z(7,10)
• W(10,6), X(6,10), Y(10,14), Z(14,10)

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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.

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Isosceles Trapezoids

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

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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.

10 1/2(3x 1 7x 1) 10 1/2(10x) 10
5x x 2
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Given isosceles trapezoid PQRS with bases PS QR
and median TV. Find the following
P
T
Q
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
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R
S
V