The document defines various types of angles and their relationships. It discusses lines, line segments, and angles. It defines acute, obtuse, right, straight, and reflex angles. It also defines complementary, supplementary, adjacent, linear pairs of angles. Examples are provided to find complementary, supplementary angles and to determine if angles form linear pairs. The document also discusses angles formed when a transversal cuts parallel lines, including corresponding, interior, exterior angles and using these properties to determine if lines are parallel.
Area of focus: Trigonometry and mathematical proofs
Topics covered:
> Trigonometry
> Right triangle definitions
> Trigonometric functions
> Special right triangles
> Law of sines
> Law of cosines
> Postulates and axioms
> Theorems
> Pythagorean Theorem
> Mathematical proof
Suggested time to complete (2 hrs):
> Teaching material (40 minutes)
> Practice activity (20 minutes)
> Final project (60 minutes)
Area of focus: Trigonometry and mathematical proofs
Topics covered:
> Trigonometry
> Right triangle definitions
> Trigonometric functions
> Special right triangles
> Law of sines
> Law of cosines
> Postulates and axioms
> Theorems
> Pythagorean Theorem
> Mathematical proof
Suggested time to complete (2 hrs):
> Teaching material (40 minutes)
> Practice activity (20 minutes)
> Final project (60 minutes)
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
The PowerPoint presentation is on the "BASICS OF TRIGONOMETRY".
It includes the --
1) Definition of Trigonometry,
2) History of Trigonometry and its Etymology,
3) Angles of a Right Triangle,
4) About different Trigonometric Ratios,
5) Some useful Mnemonics to remember the Trig. ratios,
6) Theorem, which states that --
"Trigonometric Ratios are same for the same angles"
7) Trigonometric Ratios for some specific/ standard angles.
Предлагается построить комплекс для массового производства посадочного материала тропических раков и креветок в целях реализации для дальнейшего выращивания товарной продукции заинтересованными сельхозпроизводителями
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
The PowerPoint presentation is on the "BASICS OF TRIGONOMETRY".
It includes the --
1) Definition of Trigonometry,
2) History of Trigonometry and its Etymology,
3) Angles of a Right Triangle,
4) About different Trigonometric Ratios,
5) Some useful Mnemonics to remember the Trig. ratios,
6) Theorem, which states that --
"Trigonometric Ratios are same for the same angles"
7) Trigonometric Ratios for some specific/ standard angles.
Предлагается построить комплекс для массового производства посадочного материала тропических раков и креветок в целях реализации для дальнейшего выращивания товарной продукции заинтересованными сельхозпроизводителями
Plastic Surgery Practice magazine examines the year in review in the world of Plastic Surgery, Cosmetic, and Aesthetic Medicine. #plasticsurgery #aesthetic #cosmeticsurgery #antiaging #laser
Obj. 8 Classifying Angles and Pairs of Anglessmiller5
The student will be able to (I can):
Correctly name an angle
Classify angles as acute, right, or obtuse
Identify
linear pairs
vertical angles
complementary angles
supplementary angles
and set up and solve equations.
PowerPoint presentation on the topic: Angles for year 8 students.
Presented as an online Mathematics Tutor to be selected for Mathematics position to teach year 7 to year 9 students.
As part of an online recruitment for assessment
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
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.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
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
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
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.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
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!
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
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
1. Lines and Angles
Important Definitions
Line:
A line has no thickness and has only length. A line can be extended infinitely through its both
ends.
Line segment: A line segment has a definite length, which means it has a definite
beginning point and a definite end point.
Angle – A figure formed by two rays having a common end point is called an angle. Rays are
called sides of the angle. Common end point which is shared by both the sides (rays) is called
vertex of the angle. Angles are usually measure in degree. Angle may be equal to 10 to 3600.
The symbol of angle is ∠.
Acute Angle – Angles less than 900 are called Acute Angles. For example – 200, 300, 850,
650, etc.
Obtuse Angle – Angles more than 900 are called Obtuse Angles. For example – 950, 1000,
800, 650, etc.
Right Angle – Angle equal to 900 is called Right Angle.
2. Straight Angle – Angle equal to 1800 is called Straight Angle.
Reflex Angles – Angles greater than 1800 are called reflex angles. For example 1900 is a
reflex angle.
Complementary Angles – When the sum of two angles is equal to 900, they are called
complementary angles. Both angles are called complementary to each others. For example
300 and 600, 400 and 500, 200 and 700, etc. are complementary angle since their sum is equal
to 900.
Example: Find the complementary angles for following angles:
(a) 15°
Solution: Let us assume the required complementary angle is x
So, 15° + x = 90°
Or, x = 90° - 15° = 75°
3. Hence, the required complementary angle = 75°
(b) 35°
Solution: Let us assume the required complementary angle is x
So, 35° + x = 90°
Or, x = 90° - 35° = 55°
Hence, the required complementary angle = 55°
(c) 45°
Solution: Let us assume the required complementary angle is x
So, 45° + x = 90°
Or, x = 90° - 45° = 45°
Hence, the required complementary angle = 45°
(d) 60°
Solution: Let us assume the required complementary angle is x
So, 60° + x = 90°
Or, x = 90° - 60° = 30°
Hence, the required complementary angle = 30°
(e)70°
Solution: Let us assume the required complementary angle is x
So, 70° + x = 90°
Or, x = 90° - 70° = 20°
Hence, the required complementary angle = 20°
4. Supplementary Angles - When the sum of two angles is equal to 1800, then they are called
supplementary angles. Both angles are called supplementary to each others. For example
1000 and 800, 300 and 1500, etc. are complementary angle.
Example: Find if the following angles make a pair of supplementary angles.
Solution: 60° + 120° = 180°
As the sum of these angles is equal to two right angles, so they make a pair of
supplementary angles.
Example: Find the supplementary angles for following angles:
(a)45°
Solution: Let us assume, the required angle = x
45° + x = 180°
Or, x = 180° - 45° = 135°
Hence, the required supplementary angle = 135°
(b) 25°
Solution: Let us assume, the required angle = x
25° + x = 180°
Or, x = 180° - 25° = 155°
Hence, the required supplementary angle = 155°
5. (c) 112°
Solution: Let us assume, the required angle = x
112° + x = 180°
Or, x = 180° - 112° = 68°
Hence, the required supplementary angle = 68°
(d) 130°
Solution: Let us assume, the required angle = x
130° + x = 180°
Or, x = 180° - 130° = 50°
Hence, the required supplementary angle = 50°
(e)78°
Solution: Let us assume, the required angle = x
78° + x = 180°
Or, x = 180° - 78° = 102°
Hence, the required supplementary angle = 102°
Adjacent Angles – When two angles have a common arm and common vertex and their non-
common arm are the either side of common arm, then they are called adjacent angles.
Example: Find the pairs of adjacent angles in the following figure.
Solution:
∠ 1 and ∠ 2 are adjacent to each other
∠ 2 and ∠ 3 are adjacent to each other
6. Example: Find the pairs of adjacent angles in the following figure.
Solution:
∠ APD and ∠ DPC are adjacent to each other
∠ DPC and ∠ CPB are adjacent to each other
Note: ∠ APD and ∠ CPB are not adjacent to each other, because they don’t have a common
arm in spite of having a common vertex.
Linear Pair of Angles
Two angles make a linear pair if their non-common arms are two opposite rays. In other
words, if the non-common arms of a pair of adjacent angles are in a straight line, these angles
make a linear pair.
Note: Two acute angles cannot make a linear pair because their sum will always be less than
180°. On the other hand, two right angles will always make a linear pair as their sum is equal
to 180°. It can also be said that angles of the linear pair are always supplementary to each
other.
Example: Find if following angles can make a linear pair.
7. Solution: 130° + 50° = 180°
Since the sum of these angles is equal to two right angles, so they can make a linear pair.
Example: Find if following angles can make a linear pair.
Solution: 110° + 70° = 180°
Since the sum of these angles is equal to two right angles, so they can make a linear pair.
Example: If following angles make a linear pair, find the value of q.
8. Conditions of adjacent angles –
Theyhavecommonvertex Non-common arms are the either side of common arm. Common
arm should be in the middle of rest two sides.
In the given figure ‘O’ is the common vertex and OB is the common arm. Hence, ∠a and ∠b
are called the adjacent angles.
Linear Pair - A pair of adjacent angles is called linear pair, if their two non-common arms
are opposite rays.
Here adjacent ∠A and ∠B are linear pair The angle sum of a linear pair is equal to 1800.
Vertical opposite angles: When two straight lines intersects, there are four angle formed.
The opposite angles are called vertically opposite angles. The vertically opposite angles are
equal.
In the given figure, ∠1 and ∠2,and ∠3 and ∠4 are called opposite angle.
Example: Find the values of x and y in following figure.
9. Example: An angle is equal to its complementary angle. What is the value of this angle?
Solution: Let us assume, the angle = x
Example: An angle is equal to its supplementary angle. What is the value of this angle?
Solution: Let us assume, the angle = x
Example: An angle is double its supplementary angle. What is the value of this angle?
Solution: Let us assume that the smaller angle = x, then the larger angle = 2
10. Example: An angle is double its complementary angle. What is the value of this angle?
Solution: Let us assume that the smaller angle = x, then the larger angle = 2x
Example: Find the value of x in each of the following figures.
Solution: The angles in these figures are on the same side of a line, so their sum is equal to
180°.
11. TRANSVERSAL: A line is called transversal line when it cut two lines at distinct points.
In the given picture AB is the transversal line which cut CD and EF at two different (distinct
points).
.
Transacting Lines:
When two lines pass through a common point, they are called as intersecting lines. The
common point, in this case, is called the point of intersection.
Parallel Lines: When two lines are as such that they don’t intersect anywhere; no matter in
which direction they are extended; they are called parallel lines.
Note: Two lines can be either intersecting or parallel.
Angles made by a transversal:
Understanding the concept of different angles; made by a transversal; is important for
understanding advanced concepts of geometry. Following are the names of different angles
formed when a transversal intersects two different lines.
(a) Interior angles: These are between the two lines which are being intersected by the
transversal, e.g. ∠ 3, ∠ 4, ∠ 5 and ∠ 6 are interior angles.
12. (b) Exterior angles: These are beyond the two lines which are being intersected by the
transversal, e.g. ∠ 1, ∠ 2, ∠ 7 and ∠ 8 are exterior angles.
(c) Pairs of corresponding angles: In the given figure, ∠ 1 and ∠ 5 make a pair of
corresponding angles. Similarly, other pairs of corresponding angles are; ∠ 2 and ∠ 6, ∠ 3
and ∠ 7, and ∠ 4 and ∠ 8.
(d) Pairs of alternate interior angles: In the given figure, ∠ 3 and ∠ 6 make a pair of alternate
interior angles. Similarly, ∠ 4 and ∠ 5 make another pair of alternate interior angles.
(e) Pairs of alternate exterior angles: In the given figure, ∠ 1 and ∠ 8 make a pair of alternate
exterior angles. Similarly, ∠ 2 and ∠ 7 make another pair of alternate exterior angles.
(f) Pairs of interior angles on the same side of transversal: In the given figure, ∠ 3 and ∠ 5
make a pair of interior angles on the same side of transversal. Similarly, ∠ 4 and ∠ 6 make
another pair.
Parallel Lines and transversal:
When a transversal intersects two parallel lines, then:
(a) Each pair of corresponding angles is composed of equal angles.
(b) Each pair of alternate interior angles is composed of equal angles.
(c) Each pair of alternate exterior angles is composed of equal angles.
(d) Interior angles on the same side of transversal are supplementary.
Note: The converse of above statements is also true, i.e. if a transversal intersects two given
lines at different point, then:
(a) If corresponding angles are equal, then the lines are parallel.
(b) If alternate interior angles are equal, then the lines are parallel.
(c) If alternate exterior angles are equal, then the lines are parallel.
(d) If interior angles on the same side of transversal are supplementary, then the lines are
parallel.
13. Example: If t is transversal intersecting a pair of parallel lines, find the value of x.
Solution: The known angle and the unknown angle make a pair of corresponding angles. We
know that corresponding angles are equal.
Hence, x⁰ = 127⁰
Example: If AB || CD and EF is a transversal, find the value of x.
Solution: The known angle and the unknown angle make a pair of alternate interior angles.
We know that alternate interior angles are equal.
14. Hence,
Example: If t is a transversal which intersects a pair of parallel lines, find the value of x.
Solution: The known angle and the unknown angle make a pair of alternate interior angles.
We know that alternate interior angles are equal.
Hence, x⁰ = 130⁰
Example: If ‘r’ and ‘s’ are parallel lines, find the values of ∠ 1 and ∠ 2.
Solution: The known angle and ∠ 1 make a pair of corresponding angles. We know that
corresponding angles are equal.
Hence, ∠ 1 = 76°
15. The known angle and ∠ 2 are on the same side of a line and are adjacent to each other. Hence
they are supplementary.
∠ 2 + 76° = 180°
Or, ∠ 2 = 180° - 76° = 104°
Example: It is given that DE || GB and AC is a transversal. Find the values of x, y and z in
following figure.
Solution: The known angle and ∠ z make a pair of corresponding angles and hence they are
equal.
∠ z = 125° (Corresponding angles are equal)
Now, ∠ z and ∠ y are vertically opposite angles and hence they are equal.
∠ y = ∠ z = 125° (Vertically opposite angles are equal)
Now, ∠ x and ∠ z are on the same side of a line and make a linear pair of angles. Hence,
these angles are supplementary.
∠ x + ∠ z = 180°
Or, ∠ x + 125° = 180°
Or, ∠ x = 180° - 125° = 55°
∠ x = 55°, ∠ y = 125° and ∠ z = 125°
Example: In the given figure, p || q and l is a transversal. Find the values of x and y.
16.
17. Solution:
Construction: A transversal intersects two parallel lines on two distinct points.
Given: ∠ 1 = ∠ 2
Proof: ∠ 1 = ∠ 2 (Given)
∠ 1 = ∠ 3 (Vertically opposite angles are equal)
From above equations, it is clear;
∠ 3 = ∠ 2
Let us name the angle which is vertically opposite to ∠ 3, as ∠ 4.
Proof:
∠ 1 = ∠ 2 (Given)
∠ 1 = ∠ 3 (corresponding angles are equal)
From above equations, it is clear:
∠ 2 = ∠ 3
18. ∠ 3 = ∠ 4 (Vertically opposite angles are equal.
From above equations, it is clear:
∠ 2 = ∠ 4
Solution: Let us name the angle adjacent to 120° as z.
120° + z = 180° (Linear pair of angles is supplementary)
Or, z = 180° - 120° = 60°
19. ∠ x = ∠ z = 60° (Corresponding angles are equal)
Now,
∠ x = ∠ (3y + 6) (Corresponding angles are equal)
Or, 3y + 6 = 60°
Or, 3y = 60° - 6 = 54°
Or, y = 54 ÷ 3 = 18°
Hence, x = 60° and y = 18°
Example: In the following figure, find the pair of parallel lines.
Solution: ∠ MOW ≠ ∠ MPY
So, OW and PY are not parallel
∠ MOX = 50° + 30° = 80°
∠ MOZ = 52° + 28° = 80°
So, ∠ MOX = ∠ MOZ
Since corresponding angles are equal, so OX||OZ
Example: In the following figure, a transversal is intersecting two lines at distinct
points.
20. Solution:
∠ 113° + 67° = 180°
Since internal angles on the same side of transversal are supplementary,
Example: In the given figure, a transversal is intersecting two parallel lines at distinct points.
Find the value of x.
Example: If u and v are parallel lines, find the value of x.
21. Solution: Since corresponding angles are equal
Hence, x = 53⁰
.
Exercise 1 (Based on Lines and Angles)
Question - 1. Find the complement of each of the following angles.
Answer: (a) 57° (b) 38° (c) 69° (d) 15° (e) 45° (f) 36°
22. Question - 2. Find the supplement of each of the following angles.
Answer: (a) 126.3° (b) 64.8° (c) 68.9° (d) 101.8° (e) 39.3° (f) 136.5°
Question – 3. Find the pairs of complementary and supplementary angles from these pairs of
angles.
(a) 25°, 65°
(b) 32°, 58°
(c) 109°, 71°
(d) 78°, 12°
(e) 42°, 48°
(f) 112°, 68°
(g) 128°, 52°
(h) 98°, 82°
Answer: (a) complementary (b) Complementary (c) Supplementary (d) complementary (e)
complementary (f) supplementary (g) supplementary (h) supplementary
Question - 4. Find the values of x and y in following figures:
23. Answer: (a) x = y = 60° (b) x = 81°, y = 21° (c) x = 86°, y = 69° (d) x = 33°, y = 41°
Question -5. Find the value of x in following figure:
Answer: 30°
Question – 6. Find the value of x in the following figure.
Answer: 40°
24. Question – 7 . Find the value of y in the following figure.