Post on 27-Dec-2015
UNIT 4: POLYGONS
LESSON 1: ANGLES OF POLYGONS AND
QUADRILATERALS
Diagonal: a diagonal of a polygon is a segment that connects any two nonconsecutive vertices.
Pattern Recognition:
Polygon Interior Angles Sum: The sum of the interior angle measures of an n-sided convex polygon is
(n – 2 ) *180
A decorative window is designed to have the shape of a regular octagon. Find the sum of the measures of the interior angles of the octagon.
Answer: 1080
EXTERIOR ANGLE SUM
360The sum of the exterior angles is
The measure of one exterior angle is
360
n
The measure of an interior angle of a regular polygon is 135. Find the number of sides in the polygon.
Answer: The polygon has 8 sides.
The measure of an interior angle of a regular polygon is 144. Find the number of sides in the polygon.
Answer: The polygon has 10 sides.
Find the measure of each interior angle.
Find the measure of each interior angle.
Answer:
Find the measures of an exterior angle and an interior angle of convex regular nonagon ABCDEFGHJ.
FIND X
FIND X
A. 10
B. 12
C. 14
D. 15
14
PRACTICE…..
360 36045
8n
1. Sum of the measures of the interior angles of a 11-gon is
(n – 2)180° (11 – 2)180 ° 1620
2. The measure of an exterior angle of a regular octagon is
3. The number of sides of regular polygon with exterior angle 72 ° is
4. The measure of an interior angle of a regular polygon with 30 sides
360 3605
72n n
exterior angle
15
IsoscelesTrapezoid
Quadrilaterals
Rectangle
Parallelogram
Rhombus
Square
Flow Chart
Trapezoid
PARALLELOGRAM
• 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.
Lesson 6-1: Parallelogram 16
Definition: A quadrilateral whose opposite sides are parallel.
Symbol: a smaller versionof a parallelogram
Naming:
CB
A D
PROPERTIES OF PARALLELOGRAM
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
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1. Both pairs of opposite sides are congruent.
2. Both pairs of opposite angles are congruent.
3. Consecutive angles are supplementary.
4. Diagonals bisect each other but are not congruent
P is the midpoint of .
A B
CDP
EXAMPLES
1. 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 = _____ .
18
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.)
Proving Quadrilaterals as Parallelograms
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram .
Theorem 1:
H G
E FIf one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram .
Theorem 2:
If EF GH; FG EH, then Quad. EFGH is a parallelogram.
If EF GH and EF || HG, then Quad. EFGH is a parallelogram.
Theorem:
If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
Theorem 3:
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram .
Theorem 4:
H G
EF
M
,If H F and E G
then Quad. EFGH is a parallelogram.
intIf M is the midpo of EG and FH
then Quad. EFGH is a parallelogram. EM = GM and HM = FM
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
EXAMPLE 3
A. yes
B. no
Use the Distance Formula to determine if A(3, 7), B(9, 10), C(10, 6), D(4, 3) are the vertices of a parallelogram.
Lesson 6-3: Rectangles 24
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.
25
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 26
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
EXAMPLE 5
A. x = 1
B. x = 3
C. x = 5
D. x = 10
Quadrilateral EFGH is a rectangle. If mFGE = 6x – 5 and mHFE = 4x – 5, find x.
ART Some artists stretch their own canvas over wooden frames. This allows them to customize the size of a canvas. In order to ensure that the frame is rectangular before stretching the canvas, an artist measures the sides and the diagonals of the frame. If AB = 12 inches, BC = 35 inches, CD = 12 inches, DA = 35 inches, BD = 37 inches, and AC = 37 inches, explain how an artist can be sure that the frame is rectangular.Since AB = CD, DA = BC, and AC = BD, AB CD, DA BC, and AC BD.
Answer: Because AB CD and DA BC, ABCD is a parallelogram. Since AC and BD are congruent diagonals in parallelogram ABCD, it is a rectangle.
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
≡
≡
30
PROPERTIES OF A RHOMBUS
Theorem: The diagonals of a rhombus are perpendicular.
Theorem: Each diagonal of a rhombus bisects a pair of opposite angles.
31
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
32
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.
33
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
BA
10 units 10 units
5 units
90°
45°
90°
34
CONDITIONS OF RHOMBI AND SQUARES
Theorem: If the diagonals of a parallelograms are perpendicular , then the parallelogram is a rhombus.
Theorem: If one diagonal bisects a pair of opposite angles then the parallelogram is a rhombus.
Theorem: If a quadrilateral is both a rectangle and a rhombus, then it is a square.
Theorem: If one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus.
Lesson 6-5: Trapezoid & Kites 35
TRAPEZOID
A 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 36
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 37
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