Angles
Angles are geometric figures formed by two rays with a common endpoint, known as the vertex. They are measured in degrees and can be classified based on their size, such as acute (less than 90 degrees), right (exactly 90 degrees), obtuse (more than 90 degrees but less than 180 degrees), and straight (exactly 180 degrees). Angles play a fundamental role in geometry and trigonometry.
7 Key excerpts on "Angles"
eBook - ePub- Tandi Clausen-May(Author)
- 2013(Publication Date)
- SAGE Publications Ltd(Publisher)
...C HAPTER 8 Angle Some key concepts An angle is a measure of turn, not a pair of static straight lines. A degree is just one particular fraction of a turn. Other fractions of a turn are used to measure time on an analogue clock, or movement between the points of a compass. Opposite, adjacent and alternate Angles are pairs of Angles that change together on a transforming grid. If you walk around the external Angles of a convex polygon you will always make one whole turn. If you turn through the internal Angles of a triangle you will always make a half turn – so the Angles of a triangle add up to half a turn. If you turn through the internal Angles of any quadrilateral you will always make a whole turn – so the Angles of a quadrilateral add up to a whole turn. a) What is an Angle? An angle is a measure of turn. It is a measure, not a shape. Yet it is often not classified as part of the Measures curriculum. It is commonly introduced, first and foremost, as a property of two-dimensional shapes. So – what is an angle? Can we draw an angle, and print the drawing on the page? Well … no. We can’t. An angle is a measure of turn. A turn is a movement. And we cannot draw a movement. At best, we can draw a representation of the movement – something like this, perhaps: But very often, right from the start, we speak and write of an angle, and represent it, as if it were a relationship between a pair of lines. The crucial arrow, to show that the curved line represents a movement, is lost: And in the case of a right angle, convention has done away with even the hint of movement conveyed by the arc: So here again, the predominance of print over objects and models in the representation of a mathematical concept may undermine learners’ understanding, and lead them to perceive an angle as a pair of straight lines rather than as a measure of turn...
eBook - ePub- Douglas K. Brumbaugh, Peggy L. Moch, MaryE Wilkinson(Authors)
- 2004(Publication Date)
- Routledge(Publisher)
...The most common angle found in buildings is the right angle, which has a degree measure of exactly 90° and is formed by two lines, rays, or line segments that are perpendicular (⊥) to one another. Some other Angles are described by their relation to the right angle. Angles with degree measures greater than 0° but less than 90° are called acute. Angles with degree measures greater than 90° but less than 180° are called obtuse. If two rays are joined at their endpoint and go in exactly opposite directions, then we say that they form a straight angle, which measures 180°. See Fig. 4.11 for examples of straight, obtuse, right, and acute Angles. Angles can have more than 180°, but the most commonly used ones in daily life are acute, right, obtuse, and straight. Fig. 4.11. Two of the Angles in Fig. 4.11, ∠JMI and ∠HMI, can be combined to form a right angle. When this happens, we say that the two Angles complement one another. Taken one at a time, these Angles are acute; however, if you ignore their common leg, then you can see the right angle, ∠JMH. Another special relation exists when two Angles can be combined to form a straight angle. In Fig. 4.11, ignore the common leg between ∠KMI and ∠JMI, and you can see the straight angle, ∠KMJ. We say that these two Angles supplement one another. Vertical Angles apply some of the things we are have discussed so far. Think of an X and you have two pairs of vertical Angles. As you look at that X, the top angle and the bottom angle make one pair of vertical Angles. The left and right pair of Angles formed by the X make another pair of vertical Angles. Gee, that sounds strange: Two horizontal Angles are called vertical Angles. Do not dismiss that statement too quickly, because the emphasis for vertical Angles is how they are formed. Their orientation as far as being vertical or horizontal has nothing to do with how vertical Angles are defined. Look at the top and bottom Angles in the X...
eBook - ePub- Surinder Virdi, Roy Baker, Narinder Kaur Virdi(Authors)
- 2014(Publication Date)
- Routledge(Publisher)
...CHAPTER 10 Geometry Learning outcomes: (a) Identify the different types of Angles, triAngles and quadrilaterals (b) Find Angles in triAngles, quadrilaterals and other geometrical constructions (c) Use Pythagoras’ theorem to determine diagonals in quadrilaterals and sides of right-angled triAngles (d) Calculate the circumference of a circle 10.1 Angles When two straight lines meet at a point an angle is formed, as shown in Figure 10.1. There are two ways in which an angle can be denoted, i.e. either ∠ CAB or ∠ A. Figure 10.1 The size of an angle depends on the amount of rotation between two straight lines, as illustrated in Figure 10.2. Angles are usually measured in degrees, but they can also be measured in radians. A degree, defined as of a complete revolution, is easier to understand and use as compared to the radian. Figure 10.2 shows that the rotation of line AB makes (a) revolution or 90, (b) revolution or 180, (c) revolution or270° and (d) a complete revolution or 360°. Figure 10.2 For accurate measurement of an angle a degree is further divided into minutes and seconds. There are 60 minutes in a degree and 60 seconds in a minute. This method is known as the sexagesimal system: 60 minutes (60′) = 1 degree 60 seconds (60″) = 1 minute (1′) The radian is also used as a unit for measuring Angles. The following conversion factors may be used to convert degrees into radians and vice versa. 1 radian = 57.30° (correct to 2 d.p.) π radians = 180° (π = 3.14159; correct to 5 d.p.) 2π radians = 360° Example 10.1 Convert: (a) 20°15′25″ into degrees (decimal measure) (b) 32.66° into degrees, minutes and seconds. (c) 60°25′45″ into radians. Solution: (a) The conversion of 15′25″ into degree involves two steps. The first step is to change 15′25″ into seconds, and the second to convert seconds into a degree...
- Lawrence S. Leff, Christina Pawlowski-Polanish(Authors)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...STUDY UNIT III TRIGONOMETRIC ANALYSIS 9 TRIGONOMETRY WHAT YOU WILL LEARN Trigonometry means “measurement of triAngles.” The study of trigonometry arose from the ancient need to understand the relationships between the sides and Angles of triAngles. With the development of calculus, trigonometry progressed from the study of ratios within right triAngles to trigonometric functions that could be used to better represent the circular and repeating patterns of behavior that characterize a wide range of physical phenomena in the real world. This chapter progresses from considering acute Angles in right triAngles to a more general view of Angles as rotations about the origin in the coordinate plane. By fixing the vertex of such an angle at the origin and keeping one side of the angle aligned with the positive x- axis, we can give meaning to trigonometric functions of Angles greater than 90° and less than 0°. LESSONS IN CHAPTER 9 • Lesson 9-1: Degree and Radian Measures • Lesson 9-2: Right-Triangle Trigonometry • Lesson 9-3: The General Angle • Lesson 9-4: Working with Trigonometric Functions • Lesson 9-5: Trigonometric Functions of Special Angles Lesson 9-1: Degree and Radian Measures KEY IDEAS Angle measures can be expressed in units of degrees or in real-number units called radians. Degrees are based on fractional parts of a circular revolution. Radian measure compares the length of an arc that a central angle of a circle cuts off to the radius of the circle. The Greek letter θ (theta) is commonly used to represent an angle of unknown measure. MEASURING Angles IN DEGREES AND MINUTES One degree, denoted as 1°, is of one complete revolution about a fixed point. Each of the 60 equal parts of a degree is called a minute. The notation 28°30′ is read as “28 degrees, 30 minutes.” Since 60 minutes is equivalent to 1 degree, dividing 30 minutes by 60 changes 30 minutes to a fractional part of a degree...
eBook - ePubGRE - Quantitative Reasoning
QuickStudy Laminated Reference Guide
- (Author)
- 2018(Publication Date)
- QuickStudy Reference Guides(Publisher)
...Geometry Topics include parallel and perpendicular lines, circles, triAngles, quadrilaterals, other polygons, congruent and similar figures, three-dimensional figures, area, perimeter, volume, the Pythagorean theorem, and angle measurement in degrees. Angles Supplementary Angles: The sum of the Angles equals 180°. Complementary Angles: The sum of the Angles equals 90°. Points Point: An exact position or location on a plane surface. The location of a point is determined by its ordered pair, (x, y), where x determines the horizontal location and y determines the vertical location. EX: (2, 6), (-1, -9), and (0, 0) The distance between two points is found using this formula: d = √[(x 2 – x 1) 2 + (y 2 – y 1) 2 ] EX: Find the distance between the points (-1, -1) and (2, 3) Use the distance formula: d = √[(x 2 – x 1) 2 +. (y 2 – y 1) 2 ] Plug in the known values: d = √[(2 – -1) 2 + (3 – -1) 2 ] Subtract: d = √[(3) 2 + (4) 2 ] Apply the exponents: d = √[9 + 16] Add: d = √[25] Take the square root: d = 5 Lines A line has no width or curves and continues forever in two directions. A line is the shortest distance between two points; two points define a line. The equation of a line is y = mx + b, where m represents the slope of the line and b represents the y -intercept. The slope represents the direction of the line and can be found by using the change of y divided by the change of x, or rise over run. EX: What is the slope of the line that contains the points (1, -1) and (-2, 8)? (8 – -1) ÷ (-2 – 1) = 9 ÷ -3 = -3 Lines that rise as you go to the right have a positive...
- Caryl Lorandini(Author)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...11 GEOMETRY WHAT YOU WILL LEARN • How to use facts about supplementary, complementary, vertical, and adjacent Angles to write and solve simple equations for an unknown angle • How to use facts about supplementary, vertical, and adjacent Angles to write and solve simple equations for an unknown angle when parallel lines are cut by a transversal • How to recognize and classify two-dimensional figures • How to recognize, understand, and calculate perimeter and area • How to recognize, understand, and calculate area and circumference of a circle • How to recognize and classify three-dimensional figures • How to recognize, understand, and calculate surface area and volume • How to describe the two-dimensional figures that result from slicing three-dimensional figures SECTIONS IN THIS CHAPTER • What Are Angle Pair Relationships? • What Are Vertical Angles? • What Happens When Parallel Lines Are Cut by a Transversal? • How Do We Classify Two-Dimensional Figures? • What Are Area and Perimeter and How Do We Calculate Them? • How Do We Classify Three-Dimensional Figures? • What Are Surface Area and Volume and How Do We Calculate Them? 11.1 What Are Angle Pair. Relationships? First, let’s review Angles. Angle A figure formed by two rays with a common endpoint, called the vertex of the angle. Angles are classified by their degree measure. You measure an angle with a protractor. The measure is found in degrees. There are five classifications. Acute—less than 90 degrees Right—exactly 90 degrees Obtuse—between 90 and 180 degrees Straight—exactly 180 degrees Reflex—between 180 and 360 degrees We are going to look at pairs of Angles that have specific relationships. Complementary Angles —two Angles whose sum is 90 degrees Angle The Complement 20º 70º 45º 45º 62º 28º 80º 10º 65º 25º Notice that if you add the Angles together, you will have a sum of 90 degrees...
eBook - ePubMath Dictionary for Kids
The #1 Guide for Helping Kids With Math
- Theresa R. Fitzgerald(Author)
- 2021(Publication Date)
- Routledge(Publisher)
...GEOMETRY DOI: 10.4324/9781003236443-5 Abscissa The value of the x-coordinate, or horizontal axis, on a coordinate plane or graph. It is always the first number in an ordered pair. Example: 3,2 3 is the abscissa. Adjacent To lie or stand near. Adjacent figures are near or beside each other without being separated by other figures. Adjacent Angles Two Angles that have a common vertex and one common side. Example: ∠BAK and ∠KAD are adjacent Angles. Adjoining Sharing a common boundary. Alternate Exterior Angles Exterior Angles formed by a set of parallel lines intersected by a third line. These Angles have the same measure of degrees and are the same size. Examples: ∠M and ∠D are alternate exterior Angles. ∠A and ∠N are alternate exterior Angles. Alternate Interior Angles Interior Angles formed by a set of parallel lines intersected by a third line. These Angles have the same measure of degrees and are the same size. Examples: ∠T and ∠I are alternate interior Angles. ∠F and ∠Z are alternate interior Angles. Altitude See Height. Angle The figure made by two straight lines meeting at a point (a vertex) or by two rays meeting at a point. The difference between the two lines is measured in degrees. 360° = a whole circle 180° = a half (semi) circle 90° = a quarter circle ✪ Acute Angle An angle smaller than 90 degrees. ✪ Obtuse Angle An angle larger than 90 degrees but less than 180 degrees. ✪ Right Angle An angle that has the same shape as the corner of a square. A right angle equals 90 degrees. ✪ Straight Angle An angle that measures 180 degrees. Angle of Declination An angle below the horizon, like looking down from the bow of your boat to the anchor on the lake bed. Can also be called the Angle of Depression. Angle of Inclination Angles above the horizon, like looking up from ground level to the top of something, like a building or a tree. Can also be called the Angle of Elevation. Apex An apex is the point (corner, vertex) farthest from the base of a figure...
