TechMathI - 2.7
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The document provides definitions for supplementary and complementary angles. It then gives examples of finding supplements and complements of given angle measures. It also gives word problems about finding the measures of angles that are either supplementary or complementary to each other based on additional clues about their measures.
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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.
11-4 lines and angles
The document defines an angle as the figure formed by two rays with a common endpoint called the vertex. It describes different ways to name angles and classify them as acute, right, or obtuse. Key angle relationships discussed are complementary angles, whose measures sum to 90 degrees, supplementary angles, whose measures sum to 180 degrees, vertical angles, which are congruent, and adjacent angles that form linear pairs. Examples are provided for finding missing angle measures using these relationships.
Gch04 l8
This document discusses properties of isosceles and equilateral triangles. It begins with examples calculating angle measures using properties of isosceles triangles, such as the fact that the two base angles of an isosceles triangle are congruent. It then proves theorems about isosceles and equilateral triangles, including that an equilateral triangle is also equiangular. Coordinate proofs are presented to show properties such as the midpoint of the two sides of an isosceles triangle bisecting the base. The document concludes with a lesson quiz reviewing the material.
9 trigonometric functions via the unit circle nat
The document discusses radian measurements of angles using the unit circle. It defines the unit circle as having a radius of 1 centered at the origin. The radian measurement of an angle is defined as the length of the arc cut out by that angle on the unit circle. Important conversions between degrees and radians are provided. Trigonometric functions like sine, cosine, and tangent are then defined using the unit circle for any real number angle measurement.
geometric 101.ppt
This geometry lesson covers inequalities in two triangles using the Hinge Theorem and its converse. It provides the statements of the Hinge Theorem and its converse, examples of applying these concepts to compare corresponding sides and angles in two triangles, finding ranges of possible values for variables representing angles, and a real-world application problem comparing distances traveled by two groups from a base camp. The homework assigned is selected even-numbered problems from pages 371-372 analyzing inequalities in triangles.
Gch1 l3
This document provides instruction on measuring and constructing angles:
- It defines angles and how to name them using vertex points.
- Methods for measuring angles using a protractor or transit are described. Angles can be classified as acute, right, or obtuse based on their measure.
- The angle addition and bisector properties are explained. Examples show using these concepts to find unknown angle measures.
- Exercises provide practice measuring, naming, classifying, and finding angles based on given information.
Geometry unit 12.3
This document discusses inscribed angles, which are angles whose vertex is on a circle and whose sides contain chords of the circle. It defines key terms like intercepted arc and subtend. Examples are provided to illustrate how to find the measures of inscribed angles, arcs, and angles within inscribed triangles and quadrilaterals using properties of inscribed angles. Students are then given practice problems to solve.
11.5 Special Right Triangls
The document discusses different types of triangles and their angle and side properties:
1) It explains 45-45-90 triangles, also known as isosceles right triangles, have two equal leg lengths and a hypotenuse that is √2 times the length of either leg.
2) It discusses 30-60-90 triangles, which have angles of 30, 60, and 90 degrees. The hypotenuse of each triangle is twice the length of the shorter leg, and the longer leg is √3 times the length of the shorter leg.
3) Various properties of special right triangles like the Pythagorean theorem and multiplying square roots are presented to calculate side lengths from angles or other
Geometry Section 1-5
The document defines and provides examples of angle relationships including adjacent angles, linear pairs, vertical angles, complementary angles, and supplementary angles. It defines each term and provides examples of identifying angle pairs that satisfy each relationship. It also includes examples of using properties of these angle relationships to solve problems, such as finding missing angle measures.
Trigonometry part 1
This document discusses trigonometry and trigonometric ratios. It defines trigonometry as the study of relationships between the sides and angles of a triangle. It then defines the trigonometric ratios of sine, cosine, and tangent for a right triangle in terms of the sides adjacent to, opposite to, and hypotenuse of an angle. It provides examples of using trigonometric ratios to solve for missing sides or angles of a right triangle. The document also discusses trigonometric ratios of complementary angles and provides the specific trigonometric ratios for 30, 60, 45, and 90 degree angles.
4. Quadrilateralsssssssssssssssssssssssssssssssss
1. The document discusses properties of parallelograms and provides examples of solving problems using those properties.
2. It provides two practice problems involving finding missing angle measures and side lengths of parallelograms.
3. Step-by-step solutions to the practice problems apply properties such as opposite angles being congruent and opposite sides being congruent.
Geometry unit 8.5
This document provides examples and explanations for using the Law of Sines and Law of Cosines to solve triangles. It begins with examples of finding trigonometric ratios for angles up to 180 degrees. It then shows how to use the Law of Sines to find missing side lengths or angle measures when given two angles and a side, or two sides and a non-included angle. The Law of Cosines is demonstrated for finding a missing side or angle when given two sides and the included angle, or all three sides. Multiple practice problems are provided to help understand applying these laws.
4.11.2 Special Right Triangles
1) The document discusses special right triangles, specifically 45-45-90 triangles and 30-60-90 triangles.
2) For 45-45-90 triangles, the hypotenuse is √2 times the length of either leg. For 30-60-90 triangles, the hypotenuse is 2 times the shorter leg and √3 times the longer leg.
3) Examples are provided to demonstrate using the properties of these special right triangles to calculate missing side lengths.
8.1 Special Right Triangles
1) The document discusses special right triangles, specifically 45-45-90 triangles and 30-60-90 triangles.
2) In a 45-45-90 triangle, both legs are congruent and the length of the hypotenuse is √2 times the length of either leg.
3) In a 30-60-90 triangle, the length of the hypotenuse is 2 times the length of the shorter leg, and the length of the longer leg is √3 times the length of the shorter leg.
Hprec6 4
This document provides definitions and examples regarding trigonometric functions. It defines the trig functions sine, cosine, and tangent in terms of right triangles and the unit circle. It discusses finding trig functions of angles in any quadrant using reference angles. Examples are provided to find the exact values of trig functions for quadrantal and special angles using the unit circle, as well as approximating values using a calculator for other angles. Important ideas covered include using reference angles, trig functions having exact values for quadrantal and special angles, and approximations for other angles.
7.4 Special Right Triangles
1) Special right triangles have properties that allow for determining side lengths without using the Pythagorean theorem. A 45-45-90 triangle has both legs that are congruent and the hypotenuse is √2 times the leg length. A 30-60-90 triangle has the hypotenuse that is 2 times the shorter leg and the longer leg is √3 times the shorter leg.
2) Examples are provided to demonstrate using the special right triangle properties to determine unknown side lengths.
3) Relationships between the legs and hypotenuse of 45-45-90 and 30-60-90 triangles are outlined to determine a leg from the hypotenuse or a longer leg from a shorter leg
Angles.ppt
1. The document defines key angle concepts such as acute, obtuse, right, and straight angles. It also introduces the protractor and angle addition postulate.
2. The examples demonstrate how to name angles, measure angles using a protractor, find missing angle measures using the angle addition postulate, identify congruent angles, and double an angle measure when the angle is bisected.
3. The guided practice problems apply the concepts taught in the examples to find missing angle measures, identify congruent angles, and draw and label diagrams related to bisected and straight angles.
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Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
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1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
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- 1. Bell Ringer Prove: 4x – 19 = 7x + 8, then x = -9
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