The document discusses properties of parallelograms and provides examples of determining if a quadrilateral is a parallelogram. It defines four theorems for identifying parallelograms based on opposite sides, opposite angles, bisecting diagonals, and parallel/congruent sides. Examples solve systems of equations to find values of variables such that the quadrilaterals satisfy parallelogram properties. One example uses slopes of side segments to show a quadrilateral is a parallelogram due to parallel opposite sides.
On Zα-Open Sets and Decompositions of ContinuityIJERA Editor
In this paper, we introduce and study the notion of Zα-open sets and some properties of this class of sets are investigated. Also, we introduce the class of A *L-sets via Zα-open sets. Further, by using these sets, a new decompositions of continuous functions are presented. (2000) AMS Subject Classifications: 54D10; 54C05; 54C08.
Unit 1: Topological spaces (its definition and definition of open sets)nasserfuzt
Learning Objectives:
1. To understand the definition of topology with examples
2. To know the intersection and union of topologies
3. To understand the comparison of topologies
On Zα-Open Sets and Decompositions of ContinuityIJERA Editor
In this paper, we introduce and study the notion of Zα-open sets and some properties of this class of sets are investigated. Also, we introduce the class of A *L-sets via Zα-open sets. Further, by using these sets, a new decompositions of continuous functions are presented. (2000) AMS Subject Classifications: 54D10; 54C05; 54C08.
Unit 1: Topological spaces (its definition and definition of open sets)nasserfuzt
Learning Objectives:
1. To understand the definition of topology with examples
2. To know the intersection and union of topologies
3. To understand the comparison of topologies
Contents:
1. Geometric Sequence
2. Geometric Means
3. Geometric Series
with activities
Feel free to send me a message at regie.naungayan@deped.gov.ph for corrections or other suggestions
Solution to the practice test ch 10 correlation reg ch 11 gof ch12 anovaLong Beach City College
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Elementary Statistics Practice Test 5
Module 5
Chapter 10: Correlation and Regression
Chapter 11: Goodness of Fit and Contingency Tables
Chapter 12: Analysis of Variance
Contents:
1. Geometric Sequence
2. Geometric Means
3. Geometric Series
with activities
Feel free to send me a message at regie.naungayan@deped.gov.ph for corrections or other suggestions
Solution to the practice test ch 10 correlation reg ch 11 gof ch12 anovaLong Beach City College
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Elementary Statistics Practice Test 5
Module 5
Chapter 10: Correlation and Regression
Chapter 11: Goodness of Fit and Contingency Tables
Chapter 12: Analysis of Variance
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
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We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as “distorted thinking”.
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2. Essential Questions
How do you recognize the conditions that ensure
a quadrilateral is a parallelogram?
How do you prove that a set of points forms a
parallelogram in the coordinate plane?
3. Theorems
6.9 - OPPOSITE SIDES:
6.10 - OPPOSITE ANGLES:
6.11 - DIAGONALS:
6.12 - PARALLEL CONGRUENT SET OF SIDES:
4. Theorems
6.9 - OPPOSITE SIDES: IF BOTH PAIRS OF OPPOSITE SIDES OF A
QUADRILATERAL ARE CONGRUENT, THEN THE QUADRILATERAL IS A
PARALLELOGRAM
6.10 - OPPOSITE ANGLES:
6.11 - DIAGONALS:
6.12 - PARALLEL CONGRUENT SET OF SIDES:
5. Theorems
6.9 - OPPOSITE SIDES: IF BOTH PAIRS OF OPPOSITE SIDES OF A
QUADRILATERAL ARE CONGRUENT, THEN THE QUADRILATERAL IS A
PARALLELOGRAM
6.10 - OPPOSITE ANGLES: IF BOTH PAIRS OF OPPOSITE ANGLES OF A
QUADRILATERAL ARE CONGRUENT, THEN THE QUADRILATERAL IS A
PARALLELOGRAM
6.11 - DIAGONALS:
6.12 - PARALLEL CONGRUENT SET OF SIDES:
6. Theorems
6.9 - OPPOSITE SIDES: IF BOTH PAIRS OF OPPOSITE SIDES OF A
QUADRILATERAL ARE CONGRUENT, THEN THE QUADRILATERAL IS A
PARALLELOGRAM
6.10 - OPPOSITE ANGLES: IF BOTH PAIRS OF OPPOSITE ANGLES OF A
QUADRILATERAL ARE CONGRUENT, THEN THE QUADRILATERAL IS A
PARALLELOGRAM
6.11 - DIAGONALS: IF THE DIAGONALS OF A QUADRILATERAL BISECT
EACH OTHER, THEN THE QUADRILATERAL IS A PARALLELOGRAM
6.12 - PARALLEL CONGRUENT SET OF SIDES:
7. Theorems
6.9 - OPPOSITE SIDES: IF BOTH PAIRS OF OPPOSITE SIDES OF A
QUADRILATERAL ARE CONGRUENT, THEN THE QUADRILATERAL IS A
PARALLELOGRAM
6.10 - OPPOSITE ANGLES: IF BOTH PAIRS OF OPPOSITE ANGLES OF A
QUADRILATERAL ARE CONGRUENT, THEN THE QUADRILATERAL IS A
PARALLELOGRAM
6.11 - DIAGONALS: IF THE DIAGONALS OF A QUADRILATERAL BISECT
EACH OTHER, THEN THE QUADRILATERAL IS A PARALLELOGRAM
6.12 - PARALLEL CONGRUENT SET OF SIDES: IF ONE PAIR OF
OPPOSITES SIDES OF A QUADRILATERAL IS BOTH CONGRUENT AND
PARALLEL, THEN THE QUADRILATERAL IS A PARALLELOGRAM
9. Example 1
DETERMINE WHETHER THE QUADRILATERAL IS A PARALLELOGRAM.
JUSTIFY YOUR ANSWER.
BOTH PAIRS OF OPPOSITE SIDES HAVE THE SAME MEASURE, SO
EACH OPPOSITE PAIR IS CONGRUENT, THUS MAKING IT A
PARALLELOGRAM.
10. Example 2
FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM.
11. Example 2
FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM.
4x − 1= 3(x + 2)
12. Example 2
FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM.
4x − 1= 3(x + 2)
4x − 1= 3x + 6
13. Example 2
FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM.
4x − 1= 3(x + 2)
4x − 1= 3x + 6
x = 7
14. Example 2
FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM.
4x − 1= 3(x + 2)
4x − 1= 3x + 6
x = 7
3(y + 1) = 4y − 2
15. Example 2
FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM.
4x − 1= 3(x + 2)
4x − 1= 3x + 6
x = 7
3(y + 1) = 4y − 2
3y + 3 = 4y − 2
16. Example 2
FIND X AND Y SO THAT THE QUADRILATERAL IS A PARALLELOGRAM.
4x − 1= 3(x + 2)
4x − 1= 3x + 6
x = 7
3(y + 1) = 4y − 2
3y + 3 = 4y − 2
5 = y
17. Example 3
QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND
O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO IS
A PARALLELOGRAM.
18. Example 3
QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND
O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO IS
A PARALLELOGRAM.
m(TA) =
1− 3
3 −(−1)
19. Example 3
QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND
O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO IS
A PARALLELOGRAM.
m(TA) =
1− 3
3 −(−1)
=
−2
4
20. Example 3
QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND
O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO IS
A PARALLELOGRAM.
m(TA) =
1− 3
3 −(−1)
=
−2
4
= −
1
2
21. Example 3
QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND
O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO IS
A PARALLELOGRAM.
m(TA) =
1− 3
3 −(−1)
=
−2
4
= −
1
2
m(CO) =
−1−(−3)
−2 − 2
22. Example 3
QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND
O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO IS
A PARALLELOGRAM.
m(TA) =
1− 3
3 −(−1)
=
−2
4
= −
1
2
m(CO) =
−1−(−3)
−2 − 2
=
2
−4
23. Example 3
QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND
O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO IS
A PARALLELOGRAM.
m(TA) =
1− 3
3 −(−1)
=
−2
4
= −
1
2
m(CO) =
−1−(−3)
−2 − 2
=
2
−4
= −
1
2
24. Example 3
QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND
O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO IS
A PARALLELOGRAM.
m(TA) =
1− 3
3 −(−1)
=
−2
4
= −
1
2
m(CO) =
−1−(−3)
−2 − 2
=
2
−4
= −
1
2
m(AC ) =
−3 − 1
2 − 3
25. Example 3
QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND
O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO IS
A PARALLELOGRAM.
m(TA) =
1− 3
3 −(−1)
=
−2
4
= −
1
2
m(CO) =
−1−(−3)
−2 − 2
=
2
−4
= −
1
2
m(AC ) =
−3 − 1
2 − 3
=
−4
−1
26. Example 3
QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND
O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO IS
A PARALLELOGRAM.
m(TA) =
1− 3
3 −(−1)
=
−2
4
= −
1
2
m(CO) =
−1−(−3)
−2 − 2
=
2
−4
= −
1
2
m(AC ) =
−3 − 1
2 − 3
=
−4
−1
= 4
27. Example 3
QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND
O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO IS
A PARALLELOGRAM.
m(TA) =
1− 3
3 −(−1)
=
−2
4
= −
1
2
m(CO) =
−1−(−3)
−2 − 2
=
2
−4
= −
1
2
m(AC ) =
−3 − 1
2 − 3
=
−4
−1
= 4 m(TO) =
−1− 3
−2 −(−1)
28. Example 3
QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND
O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO IS
A PARALLELOGRAM.
m(TA) =
1− 3
3 −(−1)
=
−2
4
= −
1
2
m(CO) =
−1−(−3)
−2 − 2
=
2
−4
= −
1
2
m(AC ) =
−3 − 1
2 − 3
=
−4
−1
= 4 m(TO) =
−1− 3
−2 −(−1)
=
−4
−1
29. Example 3
QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND
O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO IS
A PARALLELOGRAM.
m(TA) =
1− 3
3 −(−1)
=
−2
4
= −
1
2
m(CO) =
−1−(−3)
−2 − 2
=
2
−4
= −
1
2
m(AC ) =
−3 − 1
2 − 3
=
−4
−1
= 4 m(TO) =
−1− 3
−2 −(−1)
=
−4
−1
= 4
30. Example 3
QUADRILATERAL TACO HAS VERTICES T(−1, 3), A(3, 1), C(2, −3), AND
O(−2, −1). USE THE SLOPE FORMULA TO DETERMINE WHETHER TACO IS
A PARALLELOGRAM.
m(TA) =
1− 3
3 −(−1)
=
−2
4
= −
1
2
m(CO) =
−1−(−3)
−2 − 2
=
2
−4
= −
1
2
m(AC ) =
−3 − 1
2 − 3
=
−4
−1
= 4 m(TO) =
−1− 3
−2 −(−1)
=
−4
−1
= 4
SINCE EACH SET OF OPPOSITE SIDES HAVE THE SAME SLOPE, THEY ARE
PARALLEL. WITH EACH SET OF OPPOSITE SIDES BEING PARALLEL, TACO IS A
PARALLELOGRAM
31. Example 4
FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
32. Example 4
FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
4x − 4 = 72
33. Example 4
FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
4x − 4 = 72
4x = 76
34. Example 4
FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
4x − 4 = 72
4x = 76
x = 19
35. Example 4
FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
4x − 4 = 72
4x = 76
x = 19
180 − 72
36. Example 4
FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
4x − 4 = 72
4x = 76
x = 19
180 − 72 = 108
37. Example 4
FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
4x − 4 = 72
4x = 76
x = 19
180 − 72 = 108
8y + 8 = 108
38. Example 4
FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
4x − 4 = 72
4x = 76
x = 19
180 − 72 = 108
8y + 8 = 108
8y = 100
39. Example 4
FIND THE VALUE OF X AND Y SO THAT THE QUADRILATERAL IS A
PARALLELOGRAM.
4x − 4 = 72
4x = 76
x = 19
180 − 72 = 108
8y + 8 = 108
8y = 100
y = 12.5
