Angles

3 Key excerpts on "Angles"

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  eBook - ePub

  Spherical Geometry and Its Applications

  • Marshall A. Whittlesey(Author)
  • 2019(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  In this book we shall generally use radian measure as a default, but will also make use of degree measure when appropriate with certain applications. An angle measure given without units should be assumed to be a radian measure. If an angle is the union of two rays B A → and B C → then we use the notation ∠ ABC to denote the angle formed by these two rays, and m ∠ ABC to denote its measure. Two Angles are said to be congruent if they have the same measure. A right angle is an angle with measure π 2 radians (90 degrees). Two lines that intersect are said to be perpendicular if the Angles formed at their point of intersection are right Angles. An angle with measure less than that of a right angle is said to be acute. An angle with measure greater than that of a right angle is said to be obtuse. If the sum of the measures of two Angles equals π 2 radians (90 degrees), the two Angles are said to be complementary, and are complements of each other. If the sum of the measures of two Angles equals π radians (180 degrees), the Angles are said to be supplementary, and are supplements of each other. Two Angles are said to be in a linear pair if they have the same vertex, have one side in common, and the other two sides are opposite rays. The Angles in a linear pair are supplementary. Suppose that one angle is formed by the union of two rays r 1 → and r 2 →, and another angle is formed by the union of rays r 3 → and r 4 →. If r 1 → is opposite to r 3 → and r 2 → is opposite to r 4 →, then we sav that the Angles are vertical Angles. A pair of vertical Angles must be congruent. Given a line lying in a plane, every point in the plane belongs either to the line or to one of two convex sets called a half - plane. If a point in one half-plane and another point in the other half-plane are connected with a line segment, this segment intersects the line. The line is said to be the edge of each half-plane

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  eBook - ePub

  Basic Mathematics

  • Lawrence A. Trivieri(Author)
  • 2011(Publication Date)
  • Collins Reference

    (Publisher)

  .

  An angle can be measured in either degrees or radians; degrees are most often used in basic math. The symbol ° is used to note degree measure. Hence 47° is read, “47 degrees.” Angles are measured using an instrument called a protractor. Angles can be classified using their measures, as follows.

  1. If the measure of an angle is greater than or equal to 0° but less than 90°, the angle is called an acute angle.
  2. If the measure of an angle is 90°, the angle is called a right angle.
  3. If the measure of an angle is greater than 90° but less than 180°, the angle is called an obtuse angle.
  4. If the measure of an angle is 180°, the angle is called a straight angle.

  Consider the following

  • An angle whose measure is 37° is an acute angle.
  • An angle whose measure is 139° is an obtuse angle.
  • An angle whose measure is 179° is an obtuse angle.
  • An angle whose measure is 78° is an acute angle.
  • An angle whose measure is 90° is a right angle.

  Angles can be parts of triAngles, which, as noted in Section 8.1, are polygons having three sides. PERIMETER AND AREA OF TRIAngles

  We now consider the perimeter and area formulas of triAngles. Area is measured in square units. A square unit is a square whose sides each have 1 unit.

  Consider a triangle with sides a, b , and c and height h .

  The perimeter of a triangle (see above figure) is equal to the sum of the lengths of its sides:

  P = a + b + c

   

  The area of a triangle is equal to one half the product of its base (b ) and height (h ):

  Exercise 8.13

  A triangle has a base that measures 13 cm. The other two sides measure 11 cm and 9 cm. The height of the triangle is 8 cm. Determine the perimeter and the area of the triangle.

  Solution 8.13

  The perimeter, P , is equal to the sum of the lengths of the three sides of the triangle. We have

  P = 13 cm + 11 cm + 9 cm

  = (13+11+9) cm = 3 cm

  The area A is equal to one-half the product of the base and the height. We have

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  eBook - ePub

  GED® Math Test Tutor, For the 2014 GED® Test

  • Sandra Rush(Author)
  • 2013(Publication Date)
  • Research & Education Association

    (Publisher)

  ray goes off to ∞ in only one direction. The other end has a defnite point, or endpoint.

  If we put two line segments or rays together at one of their endpoints, at a point called the vertex, we get an angle. The measure of the angle tells how wide open it is. If the angle is all the way open and forms a straight line, it is called a straight angle, and its measure is 180°. Three other types of Angles, named for their measurements, are all less than 180°.

  As you can see, if we have several Angles with the same vertex that form a straight line, their measures must add up to 180°. But if one of the adjacent Angles is obtuse, it is the only obtuse angle because it uses up more than 90° of the 180° in a straight line.

  Two Angles whose measures add up to 180° are called supplementary Angles. They don’t have to be adjacent Angles, like the ones shown in the figure above (right), they just have to add up to 180°. Similarly, two Angles whose measures add up to 90° are called complementary Angles, and again they don’t have to be adjacent. The next figure shows adjacent and nonadjacent pairs of complementary and supplementary Angles.

  Can you see that if two adjacent Angles are equal and supplementary, they must be two right (90°) Angles (and the lines are therefore perpendicular)? What would be the measures of two adjacent Angles that are equal and complementary?. The answer is 45° each because 45° + 45° = 9 0°.

  Now let’s consider other angle relationships. If two lines intersect (cross each other), they form two pairs of vertical Angles, which is the name for the Angles across from each other. Pairs of vertical Angles are equal. Suppose in the following figure that we know A is 30°. Then C is also 30° because it is a vertical angle. Can we also figure out the measures of B and D? Yes, because A and B together form a straight line, so they must add up to 180°. So B must be 150° (that is 180° –30°), and then its vertical angle, D,

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