1.5 Complementary and Supplementary Angles Dee Black
Some slides lifted from: http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=4&ved=0CEsQFjAD&url=http%3A%2F%2Fdionmath.wikispaces.com%2Ffile%2Fview%2F2.3%2BComplementary%2Band%2BSuppl.%2BAngles.ppt&ei=_wVFUbzHCa-o4AP9ooGwBQ&usg=AFQjCNF-KDyDx_yiVaUuMJMdM6yOJqHASQ&sig2=wH2TZ9xGxsHgtc4cCnn2QQ&bvm=bv.43828540,d.dmg&cad=rja
Maths (CLASS 10) Chapter Triangles PPT
thales theorem
similar triangles
phyathagoras theorem ,etc
In this ppt all theorem are proved solution are gven
there are videos also
all topic cover
Circle - Basic Introduction to circle for class 10th maths.Let's Tute
Circle - Basics Introduction to circle for class 10th students and grade x maths and mathematics.Lets tute is an online learning centre. We provide quality education for all learners and 24/7 academic guidance through E-tutoring. Our Mission- Our aspiration is to be a renowned unpaid school on Web-World
1.5 Complementary and Supplementary Angles Dee Black
Some slides lifted from: http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=4&ved=0CEsQFjAD&url=http%3A%2F%2Fdionmath.wikispaces.com%2Ffile%2Fview%2F2.3%2BComplementary%2Band%2BSuppl.%2BAngles.ppt&ei=_wVFUbzHCa-o4AP9ooGwBQ&usg=AFQjCNF-KDyDx_yiVaUuMJMdM6yOJqHASQ&sig2=wH2TZ9xGxsHgtc4cCnn2QQ&bvm=bv.43828540,d.dmg&cad=rja
Maths (CLASS 10) Chapter Triangles PPT
thales theorem
similar triangles
phyathagoras theorem ,etc
In this ppt all theorem are proved solution are gven
there are videos also
all topic cover
Circle - Basic Introduction to circle for class 10th maths.Let's Tute
Circle - Basics Introduction to circle for class 10th students and grade x maths and mathematics.Lets tute is an online learning centre. We provide quality education for all learners and 24/7 academic guidance through E-tutoring. Our Mission- Our aspiration is to be a renowned unpaid school on Web-World
Congruent triangles. Proving congruent triangles.
Disclaimer: Some parts of the presentation are obtained from various sources. Credit to the rightful owners.
"Plug into Power: The Key to Success."_CE101rey castro
Plugging into power is the key to success. By recognizing our need for power, connecting to the source of power, and activating the power within us, we can achieve our goals and make a positive impact in the world.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Safalta Digital marketing institute in Noida, provide complete applications that encompass a huge range of virtual advertising and marketing additives, which includes search engine optimization, virtual communication advertising, pay-per-click on marketing, content material advertising, internet analytics, and greater. These university courses are designed for students who possess a comprehensive understanding of virtual marketing strategies and attributes.Safalta Digital Marketing Institute in Noida is a first choice for young individuals or students who are looking to start their careers in the field of digital advertising. The institute gives specialized courses designed and certification.
for beginners, providing thorough training in areas such as SEO, digital communication marketing, and PPC training in Noida. After finishing the program, students receive the certifications recognised by top different universitie, setting a strong foundation for a successful career in digital marketing.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
Normal labor is also termed spontaneous labor, defined as the natural physiological process through which the fetus, placenta, and membranes are expelled from the uterus through the birth canal at term (37 to 42 weeks
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
2. Chapter 6: Theorem of Pythagoras And
The Right Triangle
•6.1 The Right Triangle
•6.2 The Theorem of Pythagoras
•6.3 Special Right Triangles
3. 6.1 The Right Triangle
• DABC is a right triangle, hence m ∠ABC = 900. Therefore m ∠A
and m ∠C are complementary ( figure 6.1).
figure 6.1
4. 6.1 The Right Triangle
• Now seg.BD is a perpendicular onto seg.AC (figure 6.2).
figure 6.2
5. 6.1 The Right Triangle
• Seg.BD divides DABC into two right triangles DBDC and DADB (
figure 6.2). It can be easily proven that these two triangles are
similar to the parent DABC and therefore similar to each other.
figure 6.2
6. 6.1 The Right Triangle
• Proof:
• Consider DABC and DBDC
• ∠ ABC ≅ ∠ BDC right angles and
• ∠ BCA ≅ ∠ DCB same angle
• by AA test D ABC ~ D BDC (1)
• Similarly consider D ABC and D ADB.
• ∠ ABC ≅ ∠ADB right angle
• ∠ CAB ≅ ∠ DAB same angle
• by AA test DABC ~ DADB (2)
figure 6.2
7. 6.1 The Right Triangle
• from (1) and (2)
• DABC ~ DADB ~ DBDC.
• Since DABC ~ DBDC
• and DABC ~ DADB
figure 6.2
In (A), (seg.BC) is repeated and in
(B), (seg.AB) is repeated at the means. This is
referred to as the geometric mean.
8. 6.1 The Right Triangle
• The two proportions (A) and (B) obtained by the
similarity of DADB and DBDC with the original
triangle are stated as a theorem as follows:
• If an altitude seg.BD is drawn to the hypotenuse
seg.AC of a right triangle DABC then each leg ,
• i.e. seg.AB and seg.BC is the geometric mean
between the hypotenuse and seg.DA and seg.DC
respectively ( refer figure 6.2).
• The similarity of DBDC and DADB gives the
proportion.
figure 6.2
9. 6.1 The Right Triangle
• Example 1
• Find the geometric mean between :
• a) 2 and 18
• b) 4 and 16
• c) 9 and 25
Solution: a)
10. 6.1 The Right Triangle
• Example 2
• Find x
• Solution :
• The square of the altitude to the hypotenuse is equal to the product of the
segments cut on the hypotenuse.
x2 = (12 )( 3)
x2 = 36
x = 6.
12. 6.2 The Theorem of Pythagoras
• DABC is a right triangle.
• (seg.AB) = c
• (seg.BC) = a
• (seg.CA) = b
• CD is perpendicular to AB such that
Figure 6.3
13. 6.2 The Theorem of Pythagoras
• DABC ~ DCBD
• or (BC)2 = (AB)(CD)
• a2 = (c)(x) = cx (1)
• DABC ~ DACD
Figure 6.3
• or (AC)2 = (AB)(AD)
• b2 = (c )( y) = cy (2)
• Therefore, from (1) and (2)
• a2 + b2 = cx + cy
• = c ( x + y )
• = c (c)
• = c2
• a2 + b2 = c2
The square of the hypotenuse is equal to the sum of the squares of the legs.
14. 6.2 The Theorem of Pythagoras
• Converse of Pythagoras Theorem : In a triangle if the square of the
longest side is equal to the sum of the squares of the remaining
two sides then the longest side is the hypotenuse and the angle
opposite to it, is a right angle.
• Since m∠Q = 900 ∠B is a right angle and DABC is a right triangle.
Therefore if the square of the length of the longest side is equal to
the sum of the squares of the other two sides the triangles is a
right triangle.
15. 6.2 The Theorem of Pythagoras
• Example 1
• The length of a rectangle is 4 ft. and
the breadth is 3ft. What is the length
of its diagonal?
• Solution:
• ABCD is a rectangle such that seg.AB = 4,
seg.BC= 3. m ∠ABC is 900.
• DABC is a right triangle.
(AC) 2 = (AB)2 + (BC)2
= (4)2 + (3)2
= 16 + 9
= 25
(AC) = 5 ft.
The length of the diagonal is 5 ft.
16. 6.2 The Theorem of Pythagoras
• Example 2
• A man drives south along a straight road for 17 miles. Then turns
west at right angles and drives for 24 miles where he turns north
and continue driving for 10 miles before coming to a halt. What is
the straight distance from his starting point to his terminal point ?
17. 6.2 The Theorem of Pythagoras
Example 3
D ABC is a right triangle. m ACB = 900 , seg.AQ bisects seg.BC at Q.
Prove that 4(AQ)2 = 4( AC)2 + BC2
18. 6.3 Special Right Triangles
• The 300 - 600 - 900 triangle : If the angles of a triangle are 300 ,
600 and 900 then the side opposite to 300 is half the hypotenuse
and the side opposite to 600 is
3
2
times the hypotenuse.
Figure 6.5
19. The 300 - 600 - 900 triangle
• To prove that AB = 1/2(BC) and AC=
3
2
(BC)
• but (seg. AB) = 1/2(DB) 2
In D ABC A = 900 B = 600 and C = 300
D DCB is an equilateral triangle
(seg.DC) = (seg.BC) = (seg.DB) (1)
Therefore AB = ½(BC)
21. The 450 - 450 - 900 triangle
• If the angles of a triangle are 450 - 450 - 900 then the
perpendicular sides are
2
2
times the hypotenuse.
• To prove that AB = BC =
2
2
AC
• By Pythagoras theorem
• (seg.AB)2 + (seg.BC)2 = (seg.AC)2
• (seg.AB) = (seg.BC) . DABC is isosceles.
• (seg.AB)2 + (seg.BC)2 = 2(seg.AB)2 = (seg.AC)2
In D ABC A = 450 , B = 900 and C = 450.
Figure 6.6
(AB)2=1/2(AC)2
AB =
2
2
AC 2
and
BC =
2
2
AC 2
22. Exercises:
• Example 1
LMNO is a parallelogram such that m LON = 300 and (seg.LO) = 12 cm.
If seg.MP is the perpendicular distance between seg.LM and seg.ON ,
find (seg.MP).
23. Exercises:
Since m LON = m MNP = 300
D MNP is a 300 - 600 - 900 triangle
(MP) = ½(MN)
(MN) = (LO) = 12cm
MP = 1/2 (12cm)
MP = 6cm.
24. Exercises:
• Example 2
D PQR is an acute triangle seg.PS is perpendicular to seg.QR and seg.PT
bisects QR.
Prove that (seg.PR)2 + (seg.PQ)2 = (seg.PT)2 + (seg.QT)2