The document summarizes key concepts about polygons including theorems about interior and exterior angles of polygons, definitions of different types of polygons based on number of sides, and examples calculating interior angles of polygons using the angle sum formulas. Specifically, it provides formulas for calculating the sum of interior angles and exterior angles of any polygon based on the number of sides, and examples finding interior angles of polygons like nonagons, parallelograms, and octagons.
A strictly face regular map is a k-valent plane graph on the sphere or the entire plane with faces of size a and b such that any a-gonal face is adjacent to exactly p a-gonal face and exactly q b-gonal faces. If only one of such rule is respected then we get a weak face-regular map.
We present here enumeration technique for the face regular maps that rely on polycycle and other techniques.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
bT-Locally Closed Sets and bT-Locally Continuous Functions In Supra Topologic...IOSR Journals
The aim of this paper is to introduce a decompositions namely supra bT- locally closed sets and define supra bT-locally continuous functions. This paper also discussed some of their properties.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
A strictly face regular map is a k-valent plane graph on the sphere or the entire plane with faces of size a and b such that any a-gonal face is adjacent to exactly p a-gonal face and exactly q b-gonal faces. If only one of such rule is respected then we get a weak face-regular map.
We present here enumeration technique for the face regular maps that rely on polycycle and other techniques.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
bT-Locally Closed Sets and bT-Locally Continuous Functions In Supra Topologic...IOSR Journals
The aim of this paper is to introduce a decompositions namely supra bT- locally closed sets and define supra bT-locally continuous functions. This paper also discussed some of their properties.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Chapter 3 Polygons
3.1 Definition
3.2 Terminology
3.3 Sum Of Interior Angles Of A Polygon
3.4 Sum Of Exterior Angles Of A Polygon
3.5 Diagonals in one vertex of any Polygon
3.6 Diagonals in any vertices of a Polygon
3.7 Quadrilaterals
Chapter 3 Polygons
3.1 Definition
3.2 Terminology
3.3 Sum Of Interior Angles Of A Polygon
3.4 Sum Of Exterior Angles Of A Polygon
3.5 Diagonals in one vertex of any Polygon
3.6 Diagonals in any vertices of a Polygon
3.7 Quadrilaterals
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
For more information, visit-www.vavaclasses.com
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
3. Essential Questions
How do you find and use the sum of the measures of the interior
angles of a polygon?
How do you find and use the sum of the measures of the exterior
angles of a polygon?
Tuesday, April 29, 14
7. Theorems
6.1 - Polygon Interior Angles Sum: The sum of the interior angle
measures of a convex polygon with n sides is found with the formula
S =(n−2)180
6.2 - Polygon Exterior Angles Sum:
Tuesday, April 29, 14
8. Theorems
6.1 - Polygon Interior Angles Sum: The sum of the interior angle
measures of a convex polygon with n sides is found with the formula
S =(n−2)180
6.2 - Polygon Exterior Angles Sum: The sum of the exterior angle
measures of a convex polygon, one at each vertex, is 360°
Tuesday, April 29, 14
10. Polygons and Sides
Types of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of Polygons
# sides 3 4 5 6 7
Name Triangle Quadrilateral Pentagon Hexagon Heptagon
Tuesday, April 29, 14
11. Polygons and Sides
Types of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of Polygons
# sides 3 4 5 6 7
Name Triangle Quadrilateral Pentagon Hexagon Heptagon
Tuesday, April 29, 14
12. Polygons and Sides
Types of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of Polygons
# sides 3 4 5 6 7
Name Triangle Quadrilateral Pentagon Hexagon Heptagon
Tuesday, April 29, 14
13. Polygons and Sides
Types of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of Polygons
# sides 3 4 5 6 7
Name Triangle Quadrilateral Pentagon Hexagon Heptagon
Types of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of Polygons
# sides 8 9 10 12 n
Name Octagon Nonagon Decagon Dodecagon n-gon
Tuesday, April 29, 14
14. Polygons and Sides
Types of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of Polygons
# sides 3 4 5 6 7
Name Triangle Quadrilateral Pentagon Hexagon Heptagon
Types of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of Polygons
# sides 8 9 10 12 n
Name Octagon Nonagon Decagon Dodecagon n-gon
Tuesday, April 29, 14
15. Polygons and Sides
Types of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of Polygons
# sides 3 4 5 6 7
Name Triangle Quadrilateral Pentagon Hexagon Heptagon
Types of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of PolygonsTypes of Polygons
# sides 8 9 10 12 n
Name Octagon Nonagon Decagon Dodecagon n-gon
Tuesday, April 29, 14
16. Example 1
Find the sum of the measures of the interior angles of the following.
a. Nonagon b. 17-gon
Tuesday, April 29, 14
17. Example 1
Find the sum of the measures of the interior angles of the following.
a. Nonagon b. 17-gon
S =(n−2)180
Tuesday, April 29, 14
18. Example 1
Find the sum of the measures of the interior angles of the following.
a. Nonagon b. 17-gon
S =(n−2)180
S =(9−2)180
Tuesday, April 29, 14
19. Example 1
Find the sum of the measures of the interior angles of the following.
a. Nonagon b. 17-gon
S =(n−2)180
S =(9−2)180
=(7)180
Tuesday, April 29, 14
20. Example 1
Find the sum of the measures of the interior angles of the following.
a. Nonagon b. 17-gon
S =(n−2)180
S =(9−2)180
=(7)180
=1260°
Tuesday, April 29, 14
21. Example 1
Find the sum of the measures of the interior angles of the following.
a. Nonagon b. 17-gon
S =(n−2)180
S =(9−2)180
=(7)180
=1260°
S =(n−2)180
Tuesday, April 29, 14
22. Example 1
Find the sum of the measures of the interior angles of the following.
a. Nonagon b. 17-gon
S =(n−2)180
S =(9−2)180
=(7)180
=1260°
S =(n−2)180
S =(17−2)180
Tuesday, April 29, 14
23. Example 1
Find the sum of the measures of the interior angles of the following.
a. Nonagon b. 17-gon
S =(n−2)180
S =(9−2)180
=(7)180
=1260°
S =(n−2)180
S =(17−2)180
=(15)180
Tuesday, April 29, 14
24. Example 1
Find the sum of the measures of the interior angles of the following.
a. Nonagon b. 17-gon
S =(n−2)180
S =(9−2)180
=(7)180
=1260°
S =(n−2)180
S =(17−2)180
=(15)180
= 2700°
Tuesday, April 29, 14
25. Example 2
Find the measure of each interior angle of parallelogram RSTU.
Tuesday, April 29, 14
26. Example 2
Find the measure of each interior angle of parallelogram RSTU.
S =(n−2)180
Tuesday, April 29, 14
27. Example 2
Find the measure of each interior angle of parallelogram RSTU.
S =(n−2)180
S =(4−2)180
Tuesday, April 29, 14
28. Example 2
Find the measure of each interior angle of parallelogram RSTU.
S =(n−2)180
S =(4−2)180
=(2)180
Tuesday, April 29, 14
29. Example 2
Find the measure of each interior angle of parallelogram RSTU.
S =(n−2)180
S =(4−2)180
=(2)180
=360°
Tuesday, April 29, 14
30. Example 2
Find the measure of each interior angle of parallelogram RSTU.
S =(n−2)180
S =(4−2)180
=(2)180
=360°
11x +4+5x +11x +4+5x =360
Tuesday, April 29, 14
31. Example 2
Find the measure of each interior angle of parallelogram RSTU.
S =(n−2)180
S =(4−2)180
=(2)180
=360°
11x +4+5x +11x +4+5x =360
32x +8 =360
Tuesday, April 29, 14
32. Example 2
Find the measure of each interior angle of parallelogram RSTU.
S =(n−2)180
S =(4−2)180
=(2)180
=360°
11x +4+5x +11x +4+5x =360
32x +8 =360
−8 −8
Tuesday, April 29, 14
33. Example 2
Find the measure of each interior angle of parallelogram RSTU.
S =(n−2)180
S =(4−2)180
=(2)180
=360°
11x +4+5x +11x +4+5x =360
32x +8 =360
−8 −8
32x =352
Tuesday, April 29, 14
34. Example 2
Find the measure of each interior angle of parallelogram RSTU.
S =(n−2)180
S =(4−2)180
=(2)180
=360°
11x +4+5x +11x +4+5x =360
32x +8 =360
−8 −8
32x =352
32 32
Tuesday, April 29, 14
35. Example 2
Find the measure of each interior angle of parallelogram RSTU.
S =(n−2)180
S =(4−2)180
=(2)180
=360°
11x +4+5x +11x +4+5x =360
32x +8 =360
−8 −8
32x =352
32 32
x =11
Tuesday, April 29, 14
36. Example 2
Find the measure of each interior angle of parallelogram RSTU.
S =(n−2)180
S =(4−2)180
=(2)180
=360°
11x +4+5x +11x +4+5x =360
32x +8 =360
−8 −8
32x =352
32 32
x =11
m∠R = m∠T =5(11)
Tuesday, April 29, 14
37. Example 2
Find the measure of each interior angle of parallelogram RSTU.
S =(n−2)180
S =(4−2)180
=(2)180
=360°
11x +4+5x +11x +4+5x =360
32x +8 =360
−8 −8
32x =352
32 32
x =11
m∠R = m∠T =5(11) =55°
Tuesday, April 29, 14
38. Example 2
Find the measure of each interior angle of parallelogram RSTU.
S =(n−2)180
S =(4−2)180
=(2)180
=360°
11x +4+5x +11x +4+5x =360
32x +8 =360
−8 −8
32x =352
32 32
x =11
m∠R = m∠T =5(11) =55°
m∠S = m∠U =11(11)+4
Tuesday, April 29, 14
39. Example 2
Find the measure of each interior angle of parallelogram RSTU.
S =(n−2)180
S =(4−2)180
=(2)180
=360°
11x +4+5x +11x +4+5x =360
32x +8 =360
−8 −8
32x =352
32 32
x =11
m∠R = m∠T =5(11) =55°
m∠S = m∠U =11(11)+4
=121+4
Tuesday, April 29, 14
40. Example 2
Find the measure of each interior angle of parallelogram RSTU.
S =(n−2)180
S =(4−2)180
=(2)180
=360°
11x +4+5x +11x +4+5x =360
32x +8 =360
−8 −8
32x =352
32 32
x =11
m∠R = m∠T =5(11) =55°
m∠S = m∠U =11(11)+4
=121+4 =125°
Tuesday, April 29, 14
41. Example 3
Park City Mall is designed so that eight walkways meet in a central area in
the shape of a regular octagon. Find the measure of one of the interior
angles of the octagon.
http://www.parkcitycenter.com/directory
Tuesday, April 29, 14
42. Example 3
Park City Mall is designed so that eight walkways meet in a central area in
the shape of a regular octagon. Find the measure of one of the interior
angles of the octagon.
S =(n−2)180
http://www.parkcitycenter.com/directory
Tuesday, April 29, 14
43. Example 3
Park City Mall is designed so that eight walkways meet in a central area in
the shape of a regular octagon. Find the measure of one of the interior
angles of the octagon.
S =(n−2)180
S =(8−2)180
http://www.parkcitycenter.com/directory
Tuesday, April 29, 14
44. Example 3
Park City Mall is designed so that eight walkways meet in a central area in
the shape of a regular octagon. Find the measure of one of the interior
angles of the octagon.
S =(n−2)180
S =(8−2)180
=(6)180
http://www.parkcitycenter.com/directory
Tuesday, April 29, 14
45. Example 3
Park City Mall is designed so that eight walkways meet in a central area in
the shape of a regular octagon. Find the measure of one of the interior
angles of the octagon.
S =(n−2)180
S =(8−2)180
=(6)180
=1080° http://www.parkcitycenter.com/directory
Tuesday, April 29, 14
46. Example 3
Park City Mall is designed so that eight walkways meet in a central area in
the shape of a regular octagon. Find the measure of one of the interior
angles of the octagon.
S =(n−2)180
S =(8−2)180
=(6)180
=1080°
1080°
8 http://www.parkcitycenter.com/directory
Tuesday, April 29, 14
47. Example 3
Park City Mall is designed so that eight walkways meet in a central area in
the shape of a regular octagon. Find the measure of one of the interior
angles of the octagon.
S =(n−2)180
S =(8−2)180
=(6)180
=1080°
1080°
8
=135° http://www.parkcitycenter.com/directory
Tuesday, April 29, 14
48. Example 4
The measure of an interior angle of a regular polygon is 150°. Find the
number of sides in the polygon.
Tuesday, April 29, 14
49. Example 4
The measure of an interior angle of a regular polygon is 150°. Find the
number of sides in the polygon.
150n =(n−2)180
Tuesday, April 29, 14
50. Example 4
The measure of an interior angle of a regular polygon is 150°. Find the
number of sides in the polygon.
150n =(n−2)180
150n =180n−360
Tuesday, April 29, 14
51. Example 4
The measure of an interior angle of a regular polygon is 150°. Find the
number of sides in the polygon.
150n =(n−2)180
150n =180n−360
−180n −180n
Tuesday, April 29, 14
52. Example 4
The measure of an interior angle of a regular polygon is 150°. Find the
number of sides in the polygon.
150n =(n−2)180
150n =180n−360
−180n −180n
−30n = −360
Tuesday, April 29, 14
53. Example 4
The measure of an interior angle of a regular polygon is 150°. Find the
number of sides in the polygon.
150n =(n−2)180
150n =180n−360
−180n −180n
−30n = −360
−30 −30
Tuesday, April 29, 14
54. Example 4
The measure of an interior angle of a regular polygon is 150°. Find the
number of sides in the polygon.
150n =(n−2)180
150n =180n−360
−180n −180n
−30n = −360
−30 −30
n =12
Tuesday, April 29, 14
55. Example 4
The measure of an interior angle of a regular polygon is 150°. Find the
number of sides in the polygon.
150n =(n−2)180
150n =180n−360
−180n −180n
−30n = −360
−30 −30
n =12
There are 12 sides to the polygon
Tuesday, April 29, 14
56. Example 5
Find the value of x in the diagram.
m∠1=5x +5, m∠2 =5x, m∠3= 4x −6,
m∠4 =5x −5, m∠5= 4x +3, m∠6= 6x −12,
m∠7 = 2x +3
Tuesday, April 29, 14
57. Example 5
Find the value of x in the diagram.
m∠1=5x +5, m∠2 =5x, m∠3= 4x −6,
m∠4 =5x −5, m∠5= 4x +3, m∠6= 6x −12,
m∠7 = 2x +3
5x +5+5x +4x −6+5x −5+4x +3+6x −12+2x +3=360
Tuesday, April 29, 14
58. Example 5
Find the value of x in the diagram.
m∠1=5x +5, m∠2 =5x, m∠3= 4x −6,
m∠4 =5x −5, m∠5= 4x +3, m∠6= 6x −12,
m∠7 = 2x +3
5x +5+5x +4x −6+5x −5+4x +3+6x −12+2x +3=360
31x −12 =360
Tuesday, April 29, 14
59. Example 5
Find the value of x in the diagram.
m∠1=5x +5, m∠2 =5x, m∠3= 4x −6,
m∠4 =5x −5, m∠5= 4x +3, m∠6= 6x −12,
m∠7 = 2x +3
5x +5+5x +4x −6+5x −5+4x +3+6x −12+2x +3=360
31x −12 =360
31x =372
Tuesday, April 29, 14
60. Example 5
Find the value of x in the diagram.
m∠1=5x +5, m∠2 =5x, m∠3= 4x −6,
m∠4 =5x −5, m∠5= 4x +3, m∠6= 6x −12,
m∠7 = 2x +3
5x +5+5x +4x −6+5x −5+4x +3+6x −12+2x +3=360
31x −12 =360
31x =372
x =12
Tuesday, April 29, 14
