Lesson 6 Functions Also called Methods CS 1 Lesson 6 -- John Cole1.
Lesson 6-1
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Transcript of Lesson 6-1
Lesson 6-1: Parallelogram 1
Lesson 6-1
Parallelograms
Lesson 6-1: Parallelogram 2
Parallelogram
AB CD and BC AD
Definition: A quadrilateral whose opposite sides are parallel.
Symbol: a smaller versionof a parallelogram
Naming: A parallelogram is named using all four vertices. You can start from any one vertex, but you must
continue in a clockwise or counterclockwise direction. For example, the figure above can be either
ABCD or ADCB.
CB
A D
Lesson 6-1: Parallelogram 3
Properties of Parallelogram
AB CD and BC AD
A C and B D
180 180
180 180
m A m B and m A m D
m B m C and m C m D
2. Both pairs of opposite sides are congruent.
3. Both pairs of opposite angles are congruent.
4. Consecutive angles are supplementary.
5. Diagonals bisect each other but are not congruent
AP PC BP PDAC and BDP is the midpoint of .
A B
CDP
1. Both pairs of opposite sides are parallel
Lesson 6-1: Parallelogram 4
Examples1. Draw HKLP.
2. HK = _______ and HP = ________ .
3. m<K = m<______ .
4. m<L + m<______ = 180.5. If m<P = 65, then m<H = ____,m<K = ______ and m<L =____.
6. Draw the diagonals with their point of intersection labeled M.
7. If HM = 5, then ML = ____ .
8. If KM = 7, then KP = ____ .
9. If HL = 15, then ML = ____ .
10. If m<HPK = 36, then m<PKL = _____ .
H K
LP
PL KL
P
P or K
115° 65 115°
M
5 units
14 units7.5 units
36; (Alternate interior angles are congruent.)
5 ways to prove that a quadrilateral is a parallelogram.
1. Show that both pairs of opposite sides are || . [definition]
2. Show that both pairs of opposite sides are .
3. Show that one pair of opposite sides are both and || .4. Show that both pairs of opposite angles are .
5. Show that the diagonals bisect each other .
Examples ……Find the value of x and y that ensures the quadrilateral is a parallelogram.
Example 1:
6x4x+8
y+2
2y
6x = 4x+8
2x = 8
x = 4 units
2y = y+2
y = 2 unit
Example 2: Find the value of x and y that ensure the quadrilateral is a parallelogram.
120° 5y°
(2x + 8)°2x + 8 = 120
2x = 112
x = 56 units
5y + 120 = 180
5y = 60
y = 12 units
Lesson 6-3: Rectangles 7
Rectangles
Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other.
Definition: A rectangle is a parallelogram with four right angles.
A rectangle is a special type of parallelogram. Thus a rectangle has all the properties of a parallelogram.
Lesson 6-3: Rectangles 8
Properties of Rectangles
Therefore, ∆AEB, ∆BEC, ∆CED, and ∆AED are isosceles triangles.
If a parallelogram is a rectangle, then its diagonals are congruent.
E
D C
BA
Theorem:
Converse: If the diagonals of a parallelogram are congruent , then the parallelogram is a rectangle.
Lesson 6-3: Rectangles 9
Examples…….
1. If AE = 3x +2 and BE = 29, find the value of x.
2. If AC = 21, then BE = _______.
3. If m<1 = 4x and m<4 = 2x, find the value of x.
4. If m<2 = 40, find m<1, m<3, m<4, m<5 and m<6.
m<1=50, m<3=40, m<4=80, m<5=100, m<6=40
10.5 units
x = 9 units
x = 18 units
6
54
321
E
D C
BA
Lesson 6-4: Rhombus & Square 10
Rhombus
Definition: A rhombus is a parallelogram with four congruent sides.
Since a rhombus is a parallelogram the following are true: Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other
≡
≡
Lesson 6-4: Rhombus & Square 11
Properties of a Rhombus
Theorem: The diagonals of a rhombus are perpendicular.
Theorem: Each diagonal of a rhombus bisects a pair of opposite angles.
Lesson 6-4: Rhombus & Square 12
Rhombus Examples .....
Given: ABCD is a rhombus. Complete the following.
1. If AB = 9, then AD = ______.
2. If m<1 = 65, the m<2 = _____.
3. m<3 = ______.
4. If m<ADC = 80, the m<DAB = ______.
5. If m<1 = 3x -7 and m<2 = 2x +3, then x = _____.
54
3
21E
D C
BA9 units
65°
90°
100°
10
Lesson 6-4: Rhombus & Square 13
Square
Opposite sides are parallel. Four right angles. Four congruent sides. Consecutive angles are supplementary. Diagonals are congruent. Diagonals bisect each other. Diagonals are perpendicular. Each diagonal bisects a pair of opposite angles.
Definition: A square is a parallelogram with four congruent angles and four congruent sides.
Since every square is a parallelogram as well as a rhombus and rectangle, it has all the properties of these quadrilaterals.
Lesson 6-4: Rhombus & Square 14
Squares – Examples…...Given: ABCD is a square. Complete the following.
1. If AB = 10, then AD = _____ and DC = _____.
2. If CE = 5, then DE = _____.
3. m<ABC = _____.
4. m<ACD = _____.
5. m<AED = _____.
8 7 65
4321
E
D C
BA10 units 10 units
5 units
90°
45°
90°
Lesson 6-5: Trapezoid & Kites 15
TrapezoidA quadrilateral with exactly one pair of parallel sides.Definition:
BaseLeg
An Isosceles trapezoid is a trapezoid with congruent legs.
Trapezoid
The parallel sides are called bases and the non-parallel sides are called legs.
Isosceles trapezoid
Lesson 6-5: Trapezoid & Kites 16
Properties of Isosceles Trapezoid
A B and D C
2. The diagonals of an isosceles trapezoid are congruent.
1. Both pairs of base angles of an isosceles trapezoid are congruent.
A B
CD
Base Angles
AC DB
Lesson 6-5: Trapezoid & Kites 17
The median of a trapezoid is the segment that joins the midpoints of the legs.
The median of a trapezoid is parallel to the bases, and its measure is one-half the sum of the measures of the bases.
Median
1b
2b
1 2
1( )
2median b b
Median of a Trapezoid
Lesson 6-5: Trapezoid & Kites 18
IsoscelesTrapezoid
Quadrilaterals
Rectangle
Parallelogram
Rhombus
Square
Flow Chart
Trapezoid