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1.6-6.1 Polygon notes
A Polygon:
• •
•
•
Examples: Nonexamples:
Named by the letters of the vertices written in order
A polygon will be: Or:
Concave - Convex-
Regular Polygon:
1.6-6.1 Polygon notes
A Diagonal is a segment that connects any two vertices not already
connected by a side of the polygon.
To find the sum of the angles inside a polygon we can use the
following:
1.6-6.1 Polygon notes
Ex. 1: Find the sum of the interior angles of an:
A. 11-gon. B. 3x-gon
C. 15-gon
Ex. 2: Find the number of sides of a regular polygon if the measure
of an interior angle is:
A. 108 B. 170
If a polygon is convex, then the sum of the measures of the
exterior angles, one at each vertex, is __________.
1.6-6.1 Polygon notes
Ex. 1: Find the measures of an exterior angle of a:
A. regular dodecagon
B. polygon with 20 sides
Ex. 2: Find the number of sides of the regular polygon with
an exterior angle of:
A. 15
B. 60
1.6-6.1 Practice
Name:______________________________________Pd:______Date:_________
State if each shape is a polygon. If not, explain why it is not a polygon.
1. 2. 3. 4.
State if each figure is regular or irregular:
9. 10. 11. 12.
For each polygon name it by its number of sides, classify it as concave or convex,
and state if it is regular or irregular:
13. 14. 15. 16.
Find the sum of the measures of the interior angles of each convex polygon
1. 13-gon 2. heptagon
3. 24-gon 4. 4y-gon
1.6-6.1 Practice
Find the number of sides in the regular polygon given the measure of an interior angle
9. 144 10. 120
11. 150 12. 176.4
Find the measure of an exterior angle and an interior angle of the regular polygon
17. quadrilateral 18. pentagon
19. decagon 20. 18-gon
Find the number of sides of a regular polygon given an exterior angle measure
21. 7.5 22. 45
23. 22.5 24. 9
6.2
Parallelograms
Notes
Parallelogram: Properties:
Ex 1: Use parallelogram
ABCD to find x, y, and z
Ex 2: Use parallelogram
ABCD to find x and y
D C
B A
3x 21
2y + 3
45
2y + 5
35
3x - 4
20
D C
B A
Chapter 6 13 Glencoe Geometry
Copyright
©G
lencoe/M
cG
raw
-Hill
, a d
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Inc.
Study Guide and Intervention
Parallelograms
NAME ______________________________________________ DATE ____________ PERIOD _____
6-2
Sides and Angles of Parallelograms A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Here are fourimportant properties of parallelograms.
If PQRS is a parallelogram, then
The opposite sides of a PwQw > SwRw and PwSw >QwRw
parallelogram are congruent.
The opposite angles of a /P > /R and /S > /Q
parallelogram are congruent.
The consecutive angles of a /P and /S are supplementary; /S and /R are supplementary;
parallelogram are supplementary. /R and /Q are supplementary; /Q and /P are supplementary.
If a parallelogram has one right If m/P 5 90, then m/Q 5 90, m/R 5 90, and m/S 5 90.
angle, then it has four right angles.
If ABCD is a parallelogram, find a and b.
AwBw and CwDw are opposite sides, so AwBw > CwDw.
2a 5 34
a 5 17
/A and /C are opposite angles, so /A > /C.
8b 5 112
b 5 14
Find x and y in each parallelogram.
1. 2.
3. 4.
5. 6.
72x
30x 150
2y
608
558 5x8
2y8
6x8 12x8
3y83y
12
6x
8y
88
6x8
4y8
3x8
BA
D C34
2a
8b8
1128
P Q
RS
Less
on
6-2
Example
Exercises
Chapter 6 14 Glencoe Geometry
Study Guide and Intervention (continued)
Parallelograms
NAME ______________________________________________ DATE ____________ PERIOD _____
6-2C
opyrig
ht ©
Gle
ncoe/M
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raw
-Hill, a
div
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f The M
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raw
-Hill C
om
panie
s, In
c.
Diagonals of Parallelograms Two important properties of parallelograms deal with their diagonals.
If ABCD is a parallelogram, then:
The diagonals of a parallelogram bisect each other. AP 5 PC and DP 5 PB
Each diagonal separates a parallelogram nACD > nCAB and nADB > nCBD
into two congruent triangles.
Find x and y in parallelogram ABCD.
The diagonals bisect each other, so AE 5 CE and DE 5 BE.
6x 5 24 4y 5 18
x 5 4 y 5 4.5
Find x and y in each parallelogram.
1. 2. 3.
4. 5. 6.
Complete each statement about ~ABCD.
Justify your answer.
7. /BAC >
8. DwEw >
9. nADC >
10. AwDw ||
A B
CDE
x
17
4y
3x82y
12
3x
30810
y
2x8
608 4y8
4x
2y28
3x4y
812
A B
E
CD
6x
18
24
4y
A B
CD
P
Example
Exercises
6.3 Tests for Parallelograms Notes
If one of the following is true…
1.
2.
3.
4.
5.
…Then the quadrilateral is a parallelogram.
Find x and y so that the quadrilateral is a parallelogram
6y-42
4y
2x+36 6x-12
(56) ̊
(4y+4) ̊ (7x) ̊
(5y-26) ̊
6.3 Tests for Parallelograms Notes
Are the following figures parallelograms? Justify your
answer by telling why or why not.
1. 2.
3. 4.
5. 6.
82
o
82o 98
o
98o
12
12
7
7 8
8
10 10
6.4
Rectangle
Notes
Definition: Properties:
ABCD is a rectangle.
Find x and AC if
AC = 6x + 14 and BD = 9x + 5
Use the rectangle to find
x and y given:
284
443
542
2091
+=∠
+=∠
+=∠
+=∠
ym
ym
xm
xm
C
B
E
D
A 1 2
3
4
Chapter 6 29 Glencoe Geometry
Copyright
©G
lencoe/M
cG
raw
-Hill
, a d
ivis
ion o
f T
he M
cG
raw
-Hill
Com
panie
s,
Inc.
Skills Practice
Rectangles
NAME ______________________________________________ DATE ____________ PERIOD _____
6-4
ALGEBRA ABCD is a rectangle.
1. If AC 5 2x 1 13 and DB 5 4x 2 1, find x.
2. If AC 5 x 1 3 and DB 5 3x 2 19, find AC.
3. If AE 5 3x 1 3 and EC 5 5x 2 15, find AC.
4. If DE 5 6x 2 7 and AE 5 4x 1 9, find DB.
5. If m/DAC 5 2x 1 4 and m/BAC 5 3x 1 1, find x.
6. If m/BDC 5 7x 1 1 and m/ADB 5 9x 2 7, find m/BDC.
7. If m/ABD 5 x22 7 and m/CDB 5 4x 1 5, find x.
8. If m/BAC 5 x21 3 and m/CAD 5 x 1 15, find m/BAC.
PRST is a rectangle. Find each measure if m/1 5 50.
9. m/2 10. m/3
11. m/4 12. m/5
13. m/6 14. m/7
15. m/8 16. m/9
COORDINATE GEOMETRY Determine whether TUXY is a rectangle given
each set of vertices. Justify your answer.
17. T(23, 22), U(24, 2), X(2, 4), Y(3, 0)
18. T(26, 3), U(0, 6), X(2, 2), Y(24, 21)
19. T(4, 1), U(3, 21), X(23, 2), Y(22, 4)
T
12 3 4
567 8
9
S
RP
D C
BAE
Less
on
6-4
6.5
Rhombus
(Rhombi)
Notes
Definition: Properties:
Use Rhombus ABCD to
find x if: 84
126
−=∠
−=∠
xDECm
xEDCm
Use Rhombus ABCD to
find x and AC if:
AC = 5x – 4 and AE = 2x + 6
E
D C
B A
E
D C
B A
6.5
Square
Notes
Definition: Properties:
Use Square ABCD to
find x if: 234
56
+=∠
−=∠
xBCEm
xBAEm
Use Square ABCD to find
x and y if:
39
34
322
=
+=
+=∠
CD
yAB
xCEDm
E
D C
B A
E
D C
B A
6.6 Trapezoid Notes
Trapezoid –
If the legs are congruent then the trapezoid is an ______________________
Median –
Ex 1: If AB = 17 and DC = 29 Ex 2: If DC = 22 and XY = 15
Find XY Find AB
6.6 Trapezoid Notes
Ex 3: WXYZ is an Isosceles Trapezoid
Find the median, X∠ , and Y∠ .
Ex 4: For trapezoid ABCD, G and H are midpoints of the legs.
Find AB, D∠ , and B∠
Ex 5: JKLM is an Isosceles Trapezoid with median QR
Find LM, J∠ , and K∠
W X
Y Z
30
14
D C
H G
B A
54
36
J K
L M
Q R 29
24
54 ̊
6.6 Homework
������� is the median of trapezoid HJKL. Find each indicated value
1. Find MN if HJ = 32 and LK = 60
2. Find LK if HJ = 18 and MN = 28
3. Find MN if HJ + LK = 42
4. Find m∠LMN if m∠LHJ = 116
5. Find m∠JKL if HJKL is isosceles and m∠HLK = 62
6. Find HJ if MN = 5x + 6, HJ = 3x + 6, and LK = 8x
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