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Angles Understanding and Using Angles An angle is formed whenever two lines meet. The two lines segments are called the sides of the angle. The point at which they meet is called the vertex of the angle. Often to avoid confusion, angles are labeled with letters. Suppose we consider the angle with a vertex at point B. This angle can be called Ð ABC or Ð CBA. Notice that when three letters are used, the middle letter is the vertex. This angle can also be called Ð B or Ð x. Certain types of angles are commonly encountered. It is important to learn their names. An angle that measures 180° is called a straight angle. Angle ABC in the following figure is a straight angle. As we mentioned previously, this is one­half of a revolution. An angle whose measure is between 0° and 90° is called an acute angle. Ð DEF and Ð GHJ are both acute angles. An angle whose measure is between 90° and 180° is called an obtuse angle. Ð ABC and Ð JKL are both obtuse angles. EXAMPLE 1 In the following sketch, determine which angles are acute, obtuse, right, or straight angles.
Ð ABC and Ð CBD are acute angles, Ð CBE is an obtuse angle, Ð ABD and Ð DBE are right angles and
Ð ABE is a straight angle.
Angles Complementary & Supplementary Angles Two angles that have a sum of 90º are called complementary angles. We can therefore say that each angle is the complement of the other. Two angles that have a sum of 180º are called supplementary angles. In this case we say that each angle is the supplement of the other. EXAMPLE 2 Angle A measures 39º. (a) Find the complement of angle A. Complementary angles have a sum of 90º. So the complement of angle A is 90º ­ 39º = 51º (b) Find the supplement of angle A. Supplementary angles have a sum of 180º. So the supplement of angle A is 180º ­ 39º = 141º Vertical & Adjacent Angles Four angles are formed when two lines intersect. Think of how you have four angles if two straight streets intersect. The two angles that are opposite each other are called vertical angles. Vertical angles have the same measure. In the following sketch, angle x and angle z are vertical angles and they have the same measure. Also angle w and angle y are vertical angles, so they have the same measure. Now suppose we consider two angles that have a common side, such as angle w and angle x. Two angles that are formed by intersecting lines and share a common side are called adjacent angles. Adjacent angles of intersecting lines are supplementary. If we know that the measure of angle x is 120º, then we also know that the measure of angle w is 60º. EXAMPLE 3 In the following sketch, two lines intersect forming four angles. The measure of angle a is 55º. Find the measure of all the other angles. Since Ð a and Ð c are vertical angles, we know that they have the same measure. Thus we know that
Ð c measures 55º. Since Ð a and Ð b are adjacent angles of intersecting lines, we know that they are supplementary angles. Thus we know that Ð b measures 180º ­ 55º = 125º Finally Ð b and Ð d are vertical angles, so we know that they have the same measure. Thus we know that Ð d measures 125º.
Angles Perpendicular & Parallel Lines In mathematics there is a common notation for perpendicular lines. If line m is perpendicular to line n we write m ^ n. Parallel lines never meet. If line p is parallel to line q, we write p || q. One more situation that is very important in geometry involves lines and angles. A line that intersects two or more lines at different points is called a transversal. In the following figure, line m is a transversal that intersects line n and line p. Alternate interior angles are two angles that are on opposite sides of the transversal and between the other two lines. In the figure, Ð c and Ð w are alternate interior angles. Corresponding angles are two angles that are on the same side of the transversal and are both above (or both below) the other two lines. In the following figure, Ð a and Ð b are corresponding angles. The most important case occurs when the two lines cut by the transversal are parallel. We will state this as follows. Parallel Lines Cut by a Transversal. If two parallel lines are cut by a transversal, then the measures of corresponding angles are equal and the measures of alternate interior angles are equal. EXAMPLE 4 In the following figure, m || n and the measure of Ð a is 64º. Find the measures of Ð b,
Ð c, Ð d and Ð e . Ð a = Ð b = 64°
Ð b = Ð c = 64 °
Ð b = Ð d = 64 °
Ð e = 180 ° - 64 ° = 116 °. Practice problem: in the following figure, p || q and the measure of Ð x is 105º. Find the measures of Ð w, Ð y , Ð z , and Ð v .
Angles Exercises In exercises 1 – 6, two straight lines intersect at B, as shown in the following sketch. Answer each of the following, if possible. 1. Name all the acute angles. 2. Name all the obtuse angles. 3. Name two pairs of angles that have the same measure. 4. Name two pairs of angles that are supplementary. 5. Name two pairs of angles that are complementary. 6. Name two different straight angles. In exercises 7 – 16, name the measure of each angle, as shown in the following sketch. Assume that angle PQV is a straight angle. 7. Ð PQR
8. Ð PQS
9. Find the measure of Ð a
11. 12. Find the measures of Ð a , Ð b , and Ð c . 13. 14. Ð RQT
Find the measures of Ð a , Ð b , Ð c , Ð d , Ð e , Ð f , and Ð g if we know that p||q 15. 16.
10. Ð TQP