Angles in Circles
Angles in circles refer to the measurement of the angles formed by two intersecting lines within a circle. The central angle is an angle with its vertex at the center of the circle, while the inscribed angle is formed by two intersecting chords. These angles have specific relationships and properties, such as the central angle being twice the measure of the inscribed angle.
6 Key excerpts on "Angles in Circles"
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)
...Sometimes, for the sake of convenience, line segments are used as sides of an angle, but you should understand that any line segment is a part of a ray. The best way to name an angle is to use three points in order—first a point on one leg, then the vertex point, and then a point on the other leg (the vertex letter is always the middle of the three points defining the angle). The symbol for an angle looks like a tiny angle (∠) and sometimes it has a tiny arc drawn across it (). The sides of an angle are rays, so we need not worry about sketching them any specific length. This has implications for our use of the word “congruent” and the symbol that means congruent (). Congruent is a very strong word that means figures must be exactly the same shape and exactly the same size. The sides of an angle are rays, so they extend forever no matter how long we make them appear to be. This means that we need only measure the rotations of two angles to decide if they are congruent; in Fig. 4.9, ∠DEF ∠HIJ because they have the same measure, even though we have not sketched the rays to look the same. Fig. 4.9. Because the legs of an angle are rays, extending forever, an angle divides the plane into three distinct parts, the set of all points that are inside the legs, the set of all points that are on the legs, and the set of all points that are outside the legs. This may not be obvious when you look at an angle, because we draw only a tiny part of each ray. Figure 4.10 shows interior points, like P, are between the rays, exterior points, like Q, are outside the sides, and angle points, like D, E, and F, are on the legs of the angle. Fig. 4.10. Angles are commonly described by their degree measures. 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...
- Alfred Posamentier, Stephen Krulik(Authors)
- 2016(Publication Date)
- Routledge(Publisher)
...For a class that has already learned the relationship of an angle formed by two chords intersecting inside or on the circle and its intercepted arcs, there could be an interest to determine the relationship of the measure of an angle outside the circle, such as one formed by two secants, a secant and a tangent, or two tangents. The motivation to complete the knowledge of this topic can be stimulated by the following questions: Find the measure of the angle marked x in the following three circles (see figure 1.5). Figure 1.5 As you review the three questions after the class has had a chance to try to answer them, remind the class that they already know that an inscribed angle (figure 1.6) has half the measure of its intercepted arc. Then remind them that they also know that the measure of an angle formed by two chords intersecting inside the circle (figure 1.7) is one-half the sum of the measures of the intercepted arcs. However, they should be made to realize that to complete their knowledge of measuring angles related to the arcs of a circle they will have to include angles whose vertices are outside the circle. Figure 1.8 shows the types of external angles related to a circle. With the feeling that they will now acquire the knowledge to complete the sequence, students will be motivated to determine how to find the measures of these angles from the measures of their intercepted arcs. Figure 1.6 Figure 1.7 Figure 1.8 Topic: Tangent Segments to the Same Circle Materials or Equipment Needed Chalkboard or any other medium to present a problem to the class. Implementation of the Motivation Strategy Motivation can be created by a seemingly easy-to-understand problem that students may discover they are ill-equipped to. solve, thereby causing them to realize they have a void in their knowledge. However, the solution of the problem hinges on a mathematical concept that will be explored in the ensuing lesson...
eBook - ePubGRE - Quantitative Reasoning
QuickStudy Laminated Reference Guide
- (Author)
- 2018(Publication Date)
- QuickStudy Reference Guides(Publisher)
...A central angle can be between 0° and 180°, inclusive. Arc: A portion of the circumference. The arc length is the Circumference × Central Angle ÷ 360° EX: What is the arc length formed by a central angle of 90° of a circle with the area of 16π? Find the radius by using the area formula: A = π × r 2 Plug in the known values: 16 π = π × r 2 Divide both sides by π : 16 = r 2 Take the square root of both sides: r = 4 Find the circumference by using the circumference formula: C = 2 × π × r Plug in the known values: C = 2 × π × 4 Multiply: C = 8 π Arc length = Central angle ÷ 360° × C Plug in the known values: 90° ÷ 360° × 8 π Divide:.25 × 8 π Multiply: 2 π Chord: Line segment that connects two points of a circle. Diameter of a circle: Line segment passing. through the center of a circle and connecting two points. The diameter is the longest chord of a circle. The diameter of a circle is made up of two radii. Areas & Perimeters The area of a two-dimensional shape is the amount of space within the boundary of the shape. The area of a shape can be calculated using the corresponding area formula. EX: Find the area of a rectangle that has a base of 18 and a height of 5. A = b × h 18 × 5 = 90 The perimeter of a two-dimensional shape is the distance around the shape. It can be calculated using the corresponding perimeter formula. EX: Find the perimeter of a triangle that has sides of 3, 4, and 5. P = a + b + c 3 + 4 + 5 = 12 Shape Area Formula Perimeter Formula Square A = s × s = s 2 P = 4 × s Rectangle A = b × h P = 2 ×. (b + h) Triangle A = ½ × b × h P = a + b + c Parallelogram A = b × h P = 2 × (b + h) Trapezoid A = ½ × h × (b 1 + b 2) P = a + b + c + d Circle A = π × r 2 P = 2 × π × r Rhombus A = ½ × h × (b 1 + b 2) P = 4 × s Volumes & Surface...
- Rebecca Dayton(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...A diameter is the longest chord of the circle. A secant is a line that intersects the circle at two points. A tangent is a line that intersects the circle at exactly one point. Example: Identify each of the following in the given circle. A. chord B. secant C. tangent A. is a chord because it is a segment whose endpoints C and D are on the circle. B. is a secant because it is a line that intersects the circle at two points, A and B. C. is a tangent because it intersects the circle at exactly one point, F. Arcs An arc is a connected section of the circumference of a circle. Types of Arcs Minor arc measures less than 180° Major arc measures more than 180°. Semicircle measures exactly 180°. Note: A major arc and a semicircle are always names using three letters. Exercise 1 Identify each of the following in the circle to the right. 1. secant 2. chord 3. minor arc 4. major arc Angle Types A central angle of a circle is an angle whose vertex is located at the center of a circle. An inscribed angle of a circle is an angle whose vertex is on a circle and whose sides each intersect the circle at another point. Formulas Example: Given and, find m∠LMN. ∠ LMN is an inscribed angle, so Find mLN, Since, m∠LMN = 70° Example: Given, find m∠EFG. ∠EFG is a central angle, so Find, Since m∠EFG = 150° Note: If an inscribed angle intersects a semicircle, then the angle measure must be 90°, as shown in the following diagram. is a diameter, so. Then. Angles Formed by Secants and Tangents The measure of an angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside the circle is equal to half of the difference of the intercepted arcs. Example: Given, find m∠NOP. ∠ NOP is formed by a secant and a tangent, therefore, because the sum of the arcs must equal 360°. (360° − 200° − 95° = 65°) Exercise 2 1. Find m∠DGE 2. Find m∠PRQ 3. Find 4. Given, find m∠ABC Circumference and Area The circumference of a circle is the length around the outside of the circle...
- 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...
