Post on 05-Feb-2018
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Name the vertices:
Name the sides:
Name the diagonals containing G:
Name 2 consecutive ’s:
Name 2 nonconsecutive sides:
6.1 What is a Polygon?
Regular Polygon-
Polygon Formulas: (n = # of sides)
Sum of the interior angles of a polygon = 180°(n - 2)
Sum of the exterior angles of a polygon = 360°
Polygon Names
1. Find the sum of the measures of the angles of a convex polygon with 14 sides.
2. For the given regular polygon, find the measure of each of its interior angles:
a) dodecagon b) 16 – gon
3. Find the degree measure of each exterior angle of a regular polygon with 20 sides.
4. For the following measures of an angle of a regular polygon, find the number of sides.
a) 160 b) 140
Sides Name
N
3 ____________________________
4 ____________________________
5 ____________________________
6 ____________________________
7 ____________________________
8 ____________________________
9 ____________________________
10 ____________________________
12 ____________________________
C O
U
G A
R
One interior angle: 180(𝑛−2)
𝑛
One exterior angle: 360
𝑛
2
5. The sum of the interior angles of a convex polygon is 2520. Find the number of sides.
6. Find the number of sides of a regular polygon if the measure of one of its interior angles
Is three times the measure of its adjacent exterior angle.
Find the sum of the measures of the angles of a convex polygon with the given # of sides.
1. 17 2. 20 3. 12
For each of the following, the measure of one angle of a regular convex polygon is given. Find the # of sides.
4. 150 5. 120 6. 156
For each of the following, the number of sides of a regular polygon is given. Find the measure of each angle.
7. 4 8. 8 9. 10
Find the degree measure of one exterior angle for a regular polygon with the given # of sides
10. 8 11. 5 12. 13
15. The sum of the measure of the interior angles of a convex polygon is 1260. Classify the polygon.
16. The measure of one exterior angle of a regular polygon is 45. Classify the polygon.
17. Find the number of sides of a regular polygon, if the measure of one of its interior angles equals the
measure of its adjacent exterior angle.
18. Find the number of sides of a regular polygon, if the measure of one of its interior angles equals twice the
measure of the adjacent exterior angle.
19. Classify the regular polygon, if the measure of one of its interior angles equals one-half the measure of the
adjacent exterior angle.
20. If the sum of the measures of six interior angles of a heptagon is 755, what is the measure of the
remaining angle?
Tell whether each figure is a polygon. If it is a polygon, name it by the number
of its sides.
1.
2.
3.
4. For a polygon to be regular, it must be both equiangular and equilateral.
Name the only type of polygon that must be regular if it is equiangular._____________________
Tell whether each polygon is regular or irregular. Then tell whether it is concave or
convex.
5.
6.
7.
8. Find the sum of the interior angle measures of a 14-gon. ____________________
9. Find the measure of each interior angle of hexagon ABCDEF.
_________________________________________________________
_________________________________________________________
10. Find the value of n in pentagon PQRST.
________________________________________
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6.2-6.3
A Quadrilateral is any 4-sided polygon. The sum of interior angles for every quadrilateral is 360.
Kites
2 pairs of congruent
consecutive sides (unequal)
The diagonals are
perpendicular
Squares
All properties of the
parallelogram, the rectangle
and the rhombus
Trapezoids
Has only 1 pair of parallel sides
(base 1 & base 2)
Non-parallel sides are called legs
Rectangles
All properties of
parallelogram
Four right angles
Diagonals are congruent &
bisect each other
Consecutive sides are
perpendicular
Parallelograms
Opposite sides are congruent
Opposite sides are parallel
Opposite angles are congruent
Consecutive angles are
supplementary
The diagonals bisect each
other
Rhombus
All the properties of parallelogram
Four congruent sides
Consecutive sides are congruent
Diagonals bisect opposite angles
Diagonals perpendicular & bisect
each other
Example 1. m<DCA = 27˚
m<CAD = 38˚
m<ABD = 63˚
Find each
measure.
Solve for x and y.
4
E
D
C B
A
7
6
12 True/False 1. Every parallelogram is a quadrilateral.
2. Every quadrilateral is a parallelogram.
3. All angles of a parallelogram are
congruent.
4. Opposite sides of a parallelogram are
always congruent.
5. In APEX , PXAP // .
6. In CARY, AYCR .
7. In TOAD, TA and OD bisect each
other.
˚ ˚
4
Proving a Quadrilateral is a parallelogram
A quadrilateral is a parallelogram if:
1. both pairs of opposite sides are parallel (by definition)
2.
3.
4.
5.
5
6.4-6.5 Rectangles Definition: A rectangle is a quadrilateral with _______________________.
Definition: A rectangle is a parallelogram with _______________________.
To prove that a quadrilateral is a rectangle, prove that:
1) It is a quadrilateral with _______________________.
2) It is a parallelogram with ______________________.
3) It is a parallelogram with ______________________.
Which of the following quadrilaterals are rectangles? Justify your answer.
1. 2. 3.
For 4 – 10, ABCD is a parallelogram. From the information given, tell whether ABCD is a rectangle.
4. Given: ABAD
5. Given: DBAC
6. Given: BCD is a right angle.
7. Given: BDAC
8. Given: BDAC ; ADC is a right angle
9. Given: ADC BCD
10. Given: DAC BAC
11. Find x and y Given: Diagonals RP and SQ of rectangle PQRS meet at M.
If PM = x + 3y, SM = 4y – 2x and RM = 20.
Rhombus Definition: A quadrilateral is a rhombus iff _______________________.
Definition: A parallelogram is a rhombus iff _______________________.
Mark the rhombus. How many ’s?
What must be true about HBO?
Therefore, diagonals must be _____.
Theorem: A parallelogram is a rhombus iff _________________________.
A B
D C
R
H
O
M
B
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To Prove that a quadrilateral is a rhombus, prove that:
1) It is a quadrilateral with _________________________________________.
2) It is a parallelogram with ________________________________________.
3) It is a parallelogram with ________________________________________.
4) It is a parallelogram with ________________________________________.
Find all interior angles of the following rhombus.
Which of the following are rhombuses? Justify each answer.
1. 2. 3.
For 4 – 10, ABCD is a parallelogram. From the info. Given tell whether ABCD is a rhombus.
__________4. Given: ADAB
__________5. Given: DBAC
__________6. Given: BCD is a right angle
__________7. Given: BDAC
__________8. Given: BDAC ; ADC is a right angle
__________9. Given: ADC BCD
__________10. Given: DAC BAC
11. In rhombus ABCD, mABD = 3x – 5 and mBAC = 11x – 3. Find the measures of all the angles of the
rhombus.
12. In parallelogram ABCD, AB = 17x – 3, BC = 13x + 5, and CD = 4x + 23. Find the lengths of the sides of
parallelogram ABCD. What type of parallelogram is ABCD?
23
A
B
C
D
7
A parallelogram is a square iff it has one right angle and 2 adjacent sides.
A square is both a ____________________ and a ____________________.
A square has all of the properties of a _____________, ______________, and ________________.
To prove a quadrilateral is a square, prove that:
1) It is a rectangle with ________________________________________.
2) It is a rectangle with ________________________________________.
3) It is a rectangle with _________________________________________.
4) It is a rhombus with _________________________________________.
5) It is a rhombus with _________________________________________.
6) It is a parallelogram with _____________________________________.
Complete the following.
1. Every rectangle is also a ______________. 2. Every rhombus is also a ___________.
3. Every square is also a ___________, and a _________________.
4. A with diagonals is a __________ or a ___________.
5. A with diagonals is a ___________ or a ____________.
6. A whose diagonals are the bisectors of each other is a _________ or a ________.
True or False.
___________7. All rhombi are parallelograms.
___________8. Some rectangles are squares.
___________9. All parallelograms are rectangles.
___________10. Some rhombi are rectangles. _________11. All rectangles are squares.
___________12. All squares are rectangles. _________13. Some squares are rectangles.
Use square ABCD and the given information to find each value.
14. If mAEB = 3x, find x.
15. If mBAC = 9x, find x.
16. If AB = 2x + 1 and CD = 3x – 5, find BC
17. If mDAC = y and mBAC = 3x, find x and y.
18. If AB = x2 -15 and BC = 2x, find x.
A
D
B
C
E
A
D
B
C
E
8
6.6 Kites and Trapezoids
A kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other. Properties:
1. 2. 3.
Find the value(s) of the variable(s) in each kite.
4. 5. 6.
Can two angles of a kite be as follows? 7. opposite and acute 8. consecutive and obtuse
9. opposite and supplementary 10. consecutive and supplementary
11. opposite and complementary 12. consecutive and complementary
13. The perimeter of a kite is 66 cm. The length of one of its sides is 3 cm less
that twice the length of another. Find the length of each side of the kite.
Trapezoids
A trapezoid is a quadrilateral with exactly two parallel sides.
Parts of a trapezoid:
Isosceles Trapezoid: A trapezoid with congruent legs.
15y
(2x-4)
2x
(x+6)
(4x- 30)
(2y-20)
(3x+5)
y
3x
2
6xy
45 3
2 1
7
6
35
5 3
4
2
1
Base
Base
Leg Leg
Base Base
Leg
Leg
9054
9
find LK
c. RS = x + 5
HJ + LK = 4x + 6
find RS
Theorem: The base angles of an isosceles trapezoid are congruent.
Theorem: The diagonals of an isosceles trapezoid are congruent.
Every trapezoid contains two pairs of consecutive angles that are supplementary.
Example 1: Given the trapezoid HLJK. If the 65Jm and the 95Km , the
measure of angles H and L .
Example 2: Use Isosceles Trapezoid ABCD with
length of AD = BC.
Definition: Am altitude is a line segment from one vertex of one base of the trapezoid and perpendicular to the
opposite base.
Theorem: The length of the median of a trapezoid equals one-half the sum of the bases.
212
1bbm
Example 3: Find the missing measures of the given trapezoid.
a. mIRD
b. YR
c. DR
d. AC
Example 4: HJKL is an isosceles trapezoid with bases HJ and LK , and median RS . Use the given
information to solve each problem.
a. LK = 30
HJ = 42
find RS
b. RS = 17
HJ = 14
A B
C D
//
H
R
L K
S
J
I B 7
A C
D X Y R 75
3
a. mDAB = 75. Find the mADC.
b. AC = 40. Find BD.
c. If 256 xAm and
158 xBm , find the measures
of angle C and D.
10
Example 5: Find the length of each side of the isosceles trapezoid below.
6x + 5 B C
D A
5x + 12 28 - 3x
14x
11
Algebraic Formulas Used to Determine the Type of Quadrilateral To Show that a quadrilateral is a Parallelogram
Method 1: Both pairs of opposite sides are congruent (find distance)
Method 2: Both pairs of opposite sides are parallel (find slope)
Method 3: One pair of opposite sides are both parallel and congruent (find distance and slope)
To show that a quadrilateral is a Rhombus ****FIRST show that it is a parallelogram****
Method 1: All 4 sides are congruent
Method 2: Diagonals are perpendicular (find slope of diagonals)
To show that a quadrilateral is a Rectangle ****FIRST show that it is a parallelogram****
Method 1: All angles are right angles
Method 2: Diagonals are congruent (find distance of diagonals)
To show that a quadrilateral is an Isosceles Trapezoid
Graph first o Legs are congruent (find distance)
o Bases are parallel (find slope)
Diagonals are congruent
To show that a quadrilateral is a Kite
Two pairs of consecutive congruent sides that are not congruent to each other (find the distance)
Practice Determining Quadrilaterals (Most Precise Name)
1) Given coordinates of quadrilateral EFGH are E (6, 5), F (6, 11), G (14, 18), and H (14, 12)
A) Determine if it is a parallelogram by checking to see if opposite sides are parallel. (slope)
B) Determine if it is a parallelogram by checking to see if opposite sides are . (distance)
C) Determine if it is a parallelogram by checking to see if diagonals bisect each other. (midpoint)
2) Given quadrilateral ABCD with coordinates A (-1, -2), B (4, 4), C (10, -1), and D (5, -7).
Is quad ABCD a rectangle? __________ (Note: Use the slope formula.)
Slope of = _____ Slope of = _____
Slope of = _____ Slope of = _____
3) The coordinates of quadrilateral QRST are: Q (-2, -1), R (-1, 2), S (2, 3), and T (1, 0).
a) Find the slopes of the diagonals of quad QRST. Are they perpendicular? ________
Slope of = _______ Slope of = _______
b) Find the midpoints of each of the diagonals.
Midpoint of = (_____, _____) Midpoint of = (_____, _____)
Do they bisect each other? _________
Why or why not? _______________________________________
c) What are all the possible classifications for quad QRST? _________________________
d) The most precise name?_____________________________
AB CD
BC AD
QS RT
QS RT
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4) Given: A (-1,-6), B (1,-3), C (11, 1) and D (9,-2)
Show that Quad. ABCD is a parallelogram.
5) Given: EFGH is a parallelogram with E(-4,1), F(2,3), G(4,9) and H(-2,7)
Show that EFGH is a rhombus.
6) Given: RSTU is a parallelogram with R (-4, 5), S (-1, 9), T (7, 3) and U (4,-1)
Show that RSTU is a rectangle.
7) Given: Quad. ABCD with A (6,-4), B (6, 2), C (3, 2) and D (3,-4). Is this a parallelogram?
HW For #8–10, use slope, midpoint and/or the distance formulas to determine the most precise name for the
quadrilateral with the given vertices.
8) A (-4, 3), B (-4, 8), C (3, 10) D (3, 5)
9) A (-3, 7), B (1, 10), C (1, 5), D (-3, 2)
10) A (6, -5), B (3, 10), C (0, -5), D (3, -9)
11) A (-3, -3), B (3, 4), C (5, 0), D (-4, -1)
12) A (6, 0), B (0, 6), C (-6, 0), D (0, -6)