Angles in Polygons
Angles in polygons refer to the interior angles formed within a closed shape with straight sides. The sum of the interior angles in any polygon can be found using the formula (n-2) * 180 degrees, where n represents the number of sides. Understanding the relationship between the number of sides and the total interior angles is crucial in geometry and trigonometry.
8 Key excerpts on "Angles in Polygons"
- Rebecca Dayton(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...Concave A polygon is a two-dimensional closed shape with straight lines and is either convex or concave. Convex polygons are polygons whose interior angles are each less than 180°. That is, no angles point inward. Any polygon that is not convex is considered a concave polygon. Regular vs. Irregular A regular polygon is a polygon with all equal angles and sides. If a polygon does not have equal angles and sides, then it is an irregular polygon. Interior Angles of Polygons The sum of the interior angles of a polygon can be found using the following formula: (n − 2)180° where n is the number of sides in the polygon. Example: Find the sum of the interior angles of a 12-sided polygon. (n − 2)180° (12 − 2)180° (10)180° 1800° The sum of the interior angles of a 12-sided figure is 1800°. Example: Find the measure of each unknown angle, x. First, find the sum of the interior angles of the polygon. Since the polygon has 5 sides, it is a pentagon. The sum of the interior angles is determined as follows: (n − 2)180° (5 − 2)180° (3)180° 540° The sum of the interior angles of the figure is 540°. Using this sum, the following equation can be used to find the value of x : Exterior Angles of Polygons Regardless of the number of sides in the polygon, the sum of the exterior angles is always 360°. An interior angle and one of the exterior angles at the same vertex are supplementary. Example: Steven is drawing a regular octagon. What is the measure of one exterior angle of the octagon? The sum of the exterior angles of any polygon is 360°. Since Steven is drawing an octagon, the figure will have 8 exterior angles. Each exterior angle is congruent because the octagon is regular...
eBook - ePub- Tandi Clausen-May(Author)
- 2013(Publication Date)
- SAGE Publications Ltd(Publisher)
...360 has a lot of factors, including 3, 4, 5 and 6, so the external and internal angles of an equilateral triangle, a square, and a regular pentagon and hexagon are all whole numbers of degrees. Learners might be asked what other regular polygons have angles which are whole numbers of degrees. Then they could consider what the effect would be of adopting a different convention – with, say, 100 degrees in a whole turn. Which regular polygons would have whole-number angles then? PowerPoints PowerPoint 8-4, Internal Angles Part 1, gives a dynamic representation of the process of turning through the internal angles of a triangle, and suggests that learners follow this up by exploring the process with polygons with greater numbers of sides. PowerPoint 8-5, Internal Angles Part 2, suggests one way to think about the relationship between the number of sides in the polygons and the number of whole and half turns in its internal angles. Angle – Teaching Points Angle should be taught as a measure, not just as a property of shapes. The concept of angle as a measure of turn is often lost in static images on the printed page. Focusing on static angles can lead to misconceptions. Here again, learners need visual and kinaesthetic ‘pictures in the mind’ that they can use as a basis for their mathematical thinking. Activities that involve learners moving an object or themselves through common fractions of a turn will help them to recognise angle as a measure of turn. Degrees may be introduced as just another example of a fraction of a turn. Manipulating models will help learners to see the relationships between the angles on a grid. Angles in Polygons can be viewed dynamically, as a fraction of a turn. Resources on the CD Mathematical PowerPoints PP 8-1 What Is an Angle? PP...
eBook - ePub- Mel Friedman(Author)
- 2012(Publication Date)
- Research & Education Association(Publisher)
...CHAPTER 6 Geometry Topics CHAPTER 6 GEOMETRY TOPICS Plane geometry refers to two-dimensional shapes (that is, shapes that can be drawn on a sheet of paper), such as triangles, parallelograms, trapezoids, and circles. Three-dimensional objects (that is, shapes with depth) are the subjects of solid geometry. TRIANGLES A closed three-sided geometric figure is called a triangle. The points of the intersection of the sides of a triangle are called the vertices of the triangle. A side of a triangle is a line segment whose endpoints are the vertices of two angles of the triangle. The perimeter of a triangle is the sum of the measures of the sides of the triangle. An interior angle of a triangle is an angle formed by two sides and includes the third side within its collection of points. The sum of the measures of the interior angles of a triangle is 180°. A scalene triangle has no equal sides. An isosceles triangle has at least two equal sides. The third side is called the base of the triangle, and the base angles (the angles opposite the equal sides) are equal. An equilateral triangle has all three sides equal.. An equilateral triangle is also equiangular, with each angle equaling 60°. An acute triangle has three acute angles (less than 90°). An obtuse triangle has one obtuse angle (greater than 90°). A right triangle has a right angle. The side opposite the right angle in a right triangle is called the hypotenuse of the right triangle. The other two sides are called the legs (or arms) of the right triangle. By the Pythagorean Theorem, the lengths of the three sides of a right triangle are related by the formula c 2 = a 2 + b 2 where c is the hypotenuse and a and b are the other two sides (the legs). The Pythagorean Theorem is discussed in more detail in the next section. An altitude, or height, of a triangle is a line segment from a vertex of the triangle perpendicular to the opposite side...
eBook - ePubMath Extension Units
Book 2, Grades 4-5
- Judy Leimbach(Author)
- 2021(Publication Date)
- Routledge(Publisher)
...Assignment Sheet Angles, Triangles and Other Polygons DOI: 10.4324/9781003236481-15 Name_________________________________ I began this unit on_____________________________ (date) Mark off each activity after you have completed it and after it has been checked. Hand in all pages when you finish the unit. Lesson Completed Checked 1. Drawing Angles _____ _____ 2. Drawing Pairs of Angles _____ _____ 3. Types of Triangles _____ _____ 4. Triangle Experiment _____ _____ 5. Naming Triangles _____ _____ 6. Missing Angles _____ _____ 7. Regular Polygons and Angles _____ _____ 8. Lines of Symmetry _____ _____ 9. Symmetry Experiment _____ _____ 10. Rotation Symmetry and Design _____ _____ 11. Hexagon Design _____ _____ 12. Decagon Design _____ _____ I completed this unit on__________________________ (date) Vocabulary DOI: 10.4324/9781003236481-16 Line A set of points that extends in both directions infinitely. Ray A straight line extending from a point. Line Segment Part of a straight line. Angle A geometric figure formed by two rays with a common point (vertex). Acute angle An angle between 0° and 90°. Right angle An angle that is 90°. Obtuse angle An angle between 90° and 180°. Straight angle Straight angle Polygon A closed figure made of line segments. If it has equal angles and sides, it is a regular polygon. Adjacent angles Two angles with a common side and a common vertex. Linear Pair Two adjacent angles whose non-common sides form a straight line. Vertical angles Two angles formed by two intersecting lines that are not adjacent angles. Vertical angles have equal measures. Drawing Angles DOI: 10.4324/9781003236481-17 Name________________________________ Angle ABC can be written as ∠ABC, which denotes an angle with a point A on one ray, a point C on the other ray, and point B as the vertex...
eBook - ePub- Douglas K. Brumbaugh, Peggy L. Moch, MaryE Wilkinson(Authors)
- 2004(Publication Date)
- Routledge(Publisher)
...You can determine the measure of each of those eight angles after making only one measurement. Suppose you measure Angle 1 and find that it is 35°, then Angles 4, 5, and 8 also measure 35°. Angles 1 and 4 are formed by the same pair of lines, share a vertex, and are directly opposite one another across that vertex—they are called vertical angles and they have the same measure. Angles 1 and 5 have the same orientation, each is above one of the parallel lines and has the transversal as the second leg—they are called corresponding angles and they have the same measure. That accounts for four of the eight angles. For the other four, notice that Angles 1 and 2 are supplementary, if Angle 1 measures 35°, then Angle 2 must measure 145° (recall that the measures of supplementary angles add up to 180°). If Angle 2 measures 145°, so do Angles 3, 6, and 7. There you are! Eight angle measures determined by measuring only one. Fig. 5.28. Another application of angle measure occurs when we examine the interior angles of polygons. There are several ways to demonstrate that the three interior angle measures in any triangle total 180°. You might use your protractor to measure and total the interior angles of many, many triangles. You might cut out many, many triangles and tear off the vertices of each, lining them up to form a straight angle as shown in Fig. 5.29. A quick and convincing demonstration is to use dynamic geometry software such as Geometer’s Sketchpad ®. Fig. 5.29. The sum of the measures of the interior angles of any polygon can be determined by sectioning the polygon into triangles, as was done in Fig. 5.12. The measures of the interior angles of each triangle total 180°, thus you only need to know how many triangles are involved. For the pentagon in Fig. 5.12, the sum of the measures of the interior angles is 3 × 180 = 540°, whereas for the parallelogram in Fig...
- Sandra Rush(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...Chapter 7 The Shape of Things This chapter is about geometry. Every day, you see many things that have to do with geometry and you use geometric principles, even though you don’t think of them as geometry. Tires are circles, and they had better be attached at the exact center of the circle to function properly. Honeycombs are made up of hexagons (six-sided figures). Even the truss on a bridge is a trapezoid, and bridges are made up of many triangles because the triangles create rigidity. A lot of understanding geometry is knowing the words that describe a shape. Pay particular attention to the definitions in the following sections, although they are words you probably already know. Two words that pertain to all two-dimensional closed geometric figures are perimeter and area. (Closed means all the corners are connected.) The perimeter is the distance around a figure, or the sum of the lengths of all of its sides. A typical perimeter is a fence around a plot of land. Area is a term used for the space enclosed by any closed figure. It is expressed in square units (in 2, ft 2, and so forth) and is found by various formulas, some of which are on the GED ® test formula sheet. Typical areas that we see every day are a rug or a plot of land enclosed by a fence. Lines and Angles Geometric shapes have everything to do with lines and angles, so you must understand them first. Even circles, which themselves have no straight lines or angles, have straight lines and angles within them that tell, for example, the size of the circle as well as parts of the circle. A line actually goes on forever in both directions, or we say, “It goes on to infinity (∞) in both directions.” If we want to concentrate on a part of a line, we call that a line segment, and we show which line segment we mean by stating its endpoints. So if we are interested in a line that goes from the 1-inch to the 5-inch measure, we mean a 4-inch line segment...
eBook - ePubMath Dictionary for Kids
The #1 Guide for Helping Kids With Math
- Theresa R. Fitzgerald(Author)
- 2021(Publication Date)
- Routledge(Publisher)
...Also called a 7-gon. Hexagon A polygon with six sides and six angles. Also called a 6-gon. Horizontal Parallel to the horizon and perpendicular to vertical. Horizontal Symmetry See Line Symmetry. Hypotenuse The side of a right triangle that is opposite of the right angle. Icosagon A polygon with 20 sides and 20 angles. Also called a 20-gon. Image The visual picture, or likeness, of something produced by the reflection from a mirror. Inscribed Angle An angle in which all of its points lie on the circle’s circumference. Interior Inside a figure. Interior Angle An angle on the inside of a polygon that is formed by two adjacent sides of the polygon. Intersecting Lines Lines that meet at a point. Irregular Polygon A polygon in which not all of the sides are the same length, and not all angles have the same measure. See Polygon. Kite A quadrilateral with two pairs of equal sides; each pair shares a vertex. In a kite, the angles of unequal sides are equal, and the diagonals intersect at right angles. Also called a deltoid. Line A one-dimensional straight path that is endless in both directions. Linear Relating to a straight line; one-dimensional. Line Segment A portion (part) of a straight line. There is a point at each end of a line segment, and sometimes the ends will be labeled. Line Symmetry ✪ Horizontal Symmetry Parallel to the horizon. ✪ Vertical Symmetry Perpendicular to the horizon. Midpoint A point that is an equal distance from the two endpoints of a line segment. Net Nets are a two-dimensional representation of the faces of a three-dimensional shape. If a shape figure could be opened and laid out flat, the net is the pattern those faces would take. For some shapes there may be more than one net. Nonagon A polygon with nine sides and nine angles. Also called a 9-gon. Noncollinear Points Three points that do not lie in the same straight line. Number Pair A pair of numbers that is used to give the location of a point on a graph...
eBook - ePub- Surinder Virdi, Roy Baker, Narinder Kaur Virdi(Authors)
- 2014(Publication Date)
- Routledge(Publisher)
...Plane figures have only two dimensions, i.e. length and width. Shapes or figures made by straight lines are also called polygons, some of which are: triangles, rectangles, squares, trapeziums and pentagons. 10.3 Triangles Triangles are plane figures bounded by three straight lines. A triangle is a very stable geometric shape and it is not possible to distort it without changing the length of one or more sides. Roof trusses and trussed rafters are used in the construction of roofs to dispose of rainwater and snow quickly. Their triangular shape also provides stability, which is one of the important requirements of a roof structure. Triangles also find use in the construction of multi-storey steel frames. Triangulation, a process used to convert the rectangular grids of steel frames into triangles, is necessary to make the frames and hence the buildings more stable. 10.3.1 Types of triangle (Δ) The conventional method of denoting the angles of a triangle is to use capital letters, e.g. A, B, C, P, R, etc. The sides are given small letters; side a opposite angle A, side b opposite angle B and so on, as illustrated in Figure 10.5. Figure 10.5 Figure 10.5 shows a triangle which has each of its angles as less than 90°. This type of triangle is known as an acute angled triangle. Other types of triangles are described below: obtuse angled triangle: has one angle more than 90° (Figure 10.6a) right-angled triangle: one of the angles is equal to 90° (Figure 10.6b) equilateral triangle: has equal sides (Figure 10.6c). Because of the equal sides the angles are equal as well, each being 60°. isosceles triangle: has two sides and two equal angles (Figure 10.6d). scalene triangle: has all angles of different magnitude and all sides of different length...
