Storyboard for Trig Ratios Lesson
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The document discusses properties of right triangles and trigonometry including: 1) Identifying right triangles and their hypotenuse and legs. 2) Relationships between side lengths and angles, including using the Pythagorean theorem. 3) Defining trigonometric ratios of sine, cosine, and tangent and evaluating them given angle measures and side lengths. 4) Using trigonometry to solve real world problems involving right triangles.
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The document discusses several theorems and properties related to circles:
1) Tangent segments drawn from the same point outside a circle to the circle are equal in length.
2) Parallel chords in a circle cut congruent arcs, and congruent chords are equidistant from the circle's center.
3) When two chords intersect, the product of one chord's segments equals the product of the other's segments.
Geometric Proofs
The document discusses different types of proofs in geometry using concepts like congruence, similarity, triangles, and parallel lines. It provides examples and definitions of key terms and asks multiple choice questions to test understanding. To prove congruence, one must observe that all internal angles are the same and all angles and sides correspond. Similar shapes differ in that their lengths may be different, while congruent shapes have exactly matching side lengths and angles.
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This document provides 24 examples of different types of triangles classified by their angles and sides. The examples show triangles as scalene, acute, obtuse, isosceles, equilateral, and right triangles with the side lengths and angle measures labeled using numbers, variables, or a combination of both.
Math 7 geometry 04 angles, parallel lines, and transversals - grade 7
The document discusses angles and parallel lines. It defines parallel lines and transversals, and explains that when a transversal intersects parallel lines, it forms eight angles that can be classified as vertical, corresponding, alternate interior, alternate exterior, adjacent, consecutive interior, and consecutive exterior angles. The document states that if two parallel lines are cut by a transversal, vertical angles, corresponding angles, alternate interior angles, and alternate exterior angles are congruent, while adjacent angles, consecutive interior angles, and consecutive exterior angles are supplementary. An example problem demonstrates finding angle measures given one is known.
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This document discusses angle relationships when parallel lines are cut by a transversal. It defines key terms like alternate interior angles, corresponding angles, and vertical angles. It explains that alternate interior angles, corresponding angles, and vertical angles are congruent, while same-side interior and same-side exterior angles are supplementary. The document provides examples of solving for missing angle measures using these properties and asks students to identify angle relationships in diagrams.
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The document provides definitions and examples of different types of angles formed when a transversal line crosses two or more parallel lines, including alternate exterior angles, alternate interior angles, corresponding angles, same-sided interior angles, same-sided exterior angles, and vertical angles. Students act as vocabulary detectives to find and leave clues about the definitions of each term posted around the room. As homework, students are asked to complete a handout defining and practicing identifying the different types of angles.
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To find a side length in a right angled triangle, you need to be given one other side length and the measure of one of the acute angles. The side lengths and angle are then related using one of the three trigonometric ratios, which allows you to calculate the unknown side.
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The document discusses several theorems and properties related to circles:
1) Tangent segments drawn from the same point outside a circle to the circle are equal in length.
2) Parallel chords in a circle cut congruent arcs, and congruent chords are equidistant from the circle's center.
3) When two chords intersect, the product of one chord's segments equals the product of the other's segments.
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The document discusses different types of proofs in geometry using concepts like congruence, similarity, triangles, and parallel lines. It provides examples and definitions of key terms and asks multiple choice questions to test understanding. To prove congruence, one must observe that all internal angles are the same and all angles and sides correspond. Similar shapes differ in that their lengths may be different, while congruent shapes have exactly matching side lengths and angles.
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This document provides 24 examples of different types of triangles classified by their angles and sides. The examples show triangles as scalene, acute, obtuse, isosceles, equilateral, and right triangles with the side lengths and angle measures labeled using numbers, variables, or a combination of both.
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The document discusses angles and parallel lines. It defines parallel lines and transversals, and explains that when a transversal intersects parallel lines, it forms eight angles that can be classified as vertical, corresponding, alternate interior, alternate exterior, adjacent, consecutive interior, and consecutive exterior angles. The document states that if two parallel lines are cut by a transversal, vertical angles, corresponding angles, alternate interior angles, and alternate exterior angles are congruent, while adjacent angles, consecutive interior angles, and consecutive exterior angles are supplementary. An example problem demonstrates finding angle measures given one is known.
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This document discusses angle relationships when parallel lines are cut by a transversal. It defines key terms like alternate interior angles, corresponding angles, and vertical angles. It explains that alternate interior angles, corresponding angles, and vertical angles are congruent, while same-side interior and same-side exterior angles are supplementary. The document provides examples of solving for missing angle measures using these properties and asks students to identify angle relationships in diagrams.
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The document provides definitions and examples of different types of angles formed when a transversal line crosses two or more parallel lines, including alternate exterior angles, alternate interior angles, corresponding angles, same-sided interior angles, same-sided exterior angles, and vertical angles. Students act as vocabulary detectives to find and leave clues about the definitions of each term posted around the room. As homework, students are asked to complete a handout defining and practicing identifying the different types of angles.
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This document defines and describes different types of triangles:
- Equilateral triangles have three equal sides and three equal angles.
- Isosceles triangles have at least two equal sides.
- Scalene triangles have no equal sides.
- Right triangles have one 90 degree angle.
- Acute triangles have all angles less than 90 degrees.
- Obtuse triangles have one angle greater than 90 degrees.
It also describes three theorems used to prove triangle congruence: SSS (three equal sides), SAS (two equal sides and the included angle), and ASA (two equal angles and one included side).
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Trigonometry is a branch of mathematics that studies triangles and the relationships between sides and angles. It deals specifically with right triangles, which have one 90 degree angle. The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs. Trigonometry defines functions that express the ratios of sides, such as sine, cosine, and tangent.
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The document discusses several theorems and properties for proving when two lines are parallel, including:
1) If corresponding angles formed by a transversal intersecting two lines are congruent, then the lines are parallel.
2) If alternate interior angles or alternate exterior angles formed by a transversal are congruent, then the lines are parallel.
3) If consecutive interior angles formed by a transversal are supplementary, then the lines are parallel.
It also provides an example of using these properties to prove that if two boats sail at 45 degree angles to the constant wind, their paths will not cross since their paths are parallel.
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This document discusses properties of triangles, including classifications based on sides (equilateral, isosceles, scalene) and angles (acute). It outlines key properties such as: the sum of interior angles is 180 degrees; the exterior angle is equal to the sum of the two non-adjacent interior angles; and the triangle inequality stating the sum of any two sides must be greater than the third side. Congruence of triangles is also discussed, noting triangles are congruent when corresponding sides and angles are equal, and can be proven using ASA, SAS, or SSS criteria.
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1) A triangle is a three-sided polygon with three vertices and three edges.
2) Triangles can be classified based on side lengths (equilateral, isosceles, scalene) or interior angles (right, acute, obtuse).
3) The interior angles of any triangle always sum to 180 degrees. Congruent triangles have the same shape and size, while similar triangles have the same angle measures but sides proportional in length.
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1) The document discusses different ways to determine if two triangles are equal based on their side lengths and angles.
2) It establishes three general principles: if two triangles have (a) equal sides, (b) two equal sides and the angle between them, or (c) one equal side and equal angles at the ends, then their other properties (sides or angles) are also equal.
3) Examples are given to illustrate each principle and exceptions are noted where equality does not imply full equality.
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point,line,ray
This document defines and explains various geometric terms related to lines and angles:
- A line is a straight path extending indefinitely in both directions without endpoints. A line segment is a part of a line with two endpoints. A ray is a line segment extending indefinitely in one direction from an endpoint.
- An angle is formed by two rays with a common endpoint. The common endpoint is called the vertex. The rays are the arms of the angle. Angles can be acute, right, or obtuse depending on their measure.
- Pairs of angles include adjacent angles with a common vertex and ray, vertically opposite angles formed by intersecting lines, complementary angles with a sum of 90 degrees, and supplementary angles with a sum of 180
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This document discusses congruent triangles and their properties. It provides real-world and mathematical examples of congruent figures. The four main theorems for proving triangles are congruent are presented: SSS, ASA, SAS, and RHS. An example problem demonstrates using the RHS theorem to prove two triangles are congruent based on having equal bases and altitudes.
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The document discusses three theorems regarding triangle inequalities: 1) If one side is longer than another, the opposite angle is larger. 2) If one angle is larger than another, the opposite side is longer. 3) The measure of an exterior angle is greater than either non-adjacent interior angle. It also discusses the triangle inequality theorem - the sum of any two sides is greater than the third side.
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Congruent triangles
Es una presentación del Profesor Jim Smith sobre congruencia de triángulos muy didáctica. Viene en idioma inglés.
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This document discusses different ways to prove that two lines are parallel using a transversal. It states that if two lines are cut by a transversal and their corresponding angles, alternate interior angles, consecutive interior angles, or consecutive exterior angles are congruent or supplementary, then the lines are parallel. It provides examples of proving lines parallel by finding values of x that make the lines satisfy one of these conditions. Finally, it lists the different ways one can prove two lines are parallel.
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The document discusses several theorems and properties for proving when two lines are parallel, including:
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2) If alternate interior angles or alternate exterior angles formed by a transversal are congruent, then the lines are parallel.
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of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
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Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
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What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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Storyboard for Trig Ratios Lesson
- 1. So much for so little, power of the triangle… *
- 2. * *How do I know I have a triangle?
- 3. * *Is the triangle a right triangle? How do you know?
- 4. * *Which side is the hypotenuse and which ones are the legs? hypotenuse leg1 leg2
- 5. * *Which is the longest side? Can a leg be longer than the hypotenuse? Equal to hypotenuse? Explain. Can the lags be equal to each other? hypotenuse leg1 leg2
- 6. * *Which is the largest angle? Can all the angles be the same size? Can any one of them be obtuse? Explain.
- 7. * *How does the length of side relate to the size of the angle opposite it? A B C
- 8. * *Given two sides find the missing side dimension by Pythagoras Theorem A B C 5 4 b = ??
- 9. * *With reference to an angle, which side is opposite and which one is adjacent to the reference angle? C Adjacent Opposite Hypotenuse
- 10. * *Define three trigonometric ratios using fixed “hypotenuse” and relative “opposite” and “adjacent”: sine, cosine and tangent of each acute angle. Adjacent Opposite C B A
- 11. * *Evaluate trigonometric ratios given an acute angle and at least one side. C B A 55º 12 cos C = sine C= sin B = cos B = tan B = tan C =
- 12. * *Compare trig ratios of similar right triangles and make and verify a conjecture A B C 15 9 12 Q P R 5 3 4
- 13. * *Find missing angle measures and side lengths from limited information given by using Pythagorean Theorem and/or trigonometric ratios C B A 55º 12 c am B = ?? a = ?? c = ??
- 14. * *Solve real world problems using trigonometry At 57" from the base of a building you need to look up at 55° to see the top of a building. What is the height of the building?