6 1 NAMING AND CLASSIFYING ANGLES AND TRIANGLES 6 1 Naming and Classifying Angles and Triangles Points, Lines, and Rays In the world of math, it is sometimes necessary to refer to a specific point in space. Simply draw a small dot with a pencil to represent a point. A point has no size; its only function is to show position. To name a point, use a single capital letter. M Point M If you draw two points on a sheet of paper, a line can be used to connect them. Imagine this line as being perfectly straight and continuing without end in opposite directions. It has no thickness. To name a line, use any two points on the line. Line MN, or MN A ray is part of a line that extends without end in one direction. In MN, which is read as ray MN, M is the endpoint. The second point that is used to name the ray can be any point other than the endpoint. You could also name this ray MO. Ray MN, or MO 286 HotTopics
Look at the line below. 1 Name the line in two different ways. 2 What is the endpoint of KL? Naming Angles Imagine two different rays with the same endpoint. Together they form what is called an angle. The point they have in common is 3 called the vertex of the angle. The rays form the sides of the angle. The angle above is made up of QP and QR. Q is the common endpoint of the two rays. Point Q is the vertex of the angle. Instead of writing the word angle, you can use the symbol for an angle, which is. There are several ways to name an angle. You can name it using the three letters of the points that make up the two rays with the vertex as the middle letter ( PQR, or RQP). You can also use just the letter of the vertex to name the angle ( Q). Sometimes you might want to name an angle with a number ( 3). When more than one angle is formed at a vertex, you use three letters to name each of the angles. Because G is the vertex of three different angles, each angle needs three letters to name it: DGF; DGE; EGF. Naming and Classifying Angles and Triangles 287
6 1 NAMING AND CLASSIFYING ANGLES AND TRIANGLES Look at the angles formed by the rays at the right. 3 Name the vertex. 4 Name all the angles. Measuring Angles You measure an angle in degrees, using a protractor (p. 379). The number of degrees in an angle will be greater than 0 and less than or equal to 180. EXAMPLE Measuring with a Protractor Measure ABC. Place the center point of the protractor on the vertex of the angle. Align the 0 line on the protractor with one side of the angle. Read the number of degrees on the scale where it intersects the second side of the angle. So, m ABC = 135. Measure the angles, using a protractor. 5 PSQ 6 QSR 7 PSR 288 HotTopics
Classifying Angles You can classify angles by their measures. Acute angle measures less than 90 Right angle measures 90 Obtuse angle measures greater than 90 and less than 180 Straight angle measures 180 Reflex angle measures greater than 180 Angles that share a side are called adjacent angles. You can add measures if the angles are adjacent. m KNL = 25 m LNM = 65 m KNM = 25 + 65 = 90 Because the sum is 90, you know that KNM is a right angle. 25 65 Use a protractor to measure and classify each angle. 8 SQR 9 PQR 10 PQS Naming and Classifying Angles and Triangles 289
6 1 NAMING AND CLASSIFYING ANGLES AND TRIANGLES Special Pairs of Angles When the sum of two angles equals 180, they are called supplementary angles. Supplementary 70 110 Not Supplementary 50 70 ABD = 70 JGL = 50 DBC = 110 LGH = 70 70 + 110 = 180 50 + 70 = 120 ABD + DBC = 180 JGL + LGH = 120 Opposite angles formed by two intersecting lines are called vertical angles. 1 and 4 are vertical angles. 2 and 3 are vertical angles. 1 2 3 4 Identify each pair of angles as supplementary or vertical. 11 12 6 and 8 6 9 7 8 290 HotTopics
When two angles have the same angle measure, they are called congruent angles. FTS and GPQ are congruent because each angle measures 45. 45 45 If the sum of the measure of two angles is 90, then the angles are complementary angles. Complementary Not Complementary 60 30 30 30 NMO = 30 PQR = 30 LMN = 60 RQS = 30 30 + 60 = 90 30 + 30 = 60 NMO + LMN = 90 PQR + RQS = 60 Identify each pair of angles as complementary or congruent. 13 14 Naming and Classifying Angles and Triangles 291
6 1 NAMING AND CLASSIFYING ANGLES AND TRIANGLES Triangles Triangles are polygons (p. 298) that have three sides, three vertices, and three angles. You name a triangle using the three vertices in any order. ABC is read triangle ABC. Classifying Triangles Like angles, triangles are classified by their angle measures. They are also classified by the number of congruent sides, which are sides with equal length. Acute triangle three acute angles Equilateral triangle three congruent sides; three congruent angles Obtuse triangle one obtuse angle Isosceles triangle at least two congruent sides; at least two congruent angles Right triangle one right angle Scalene triangle no congruent sides The sum of the measures of the three angles in a triangle is always 180. 60 70 50 In DEF, m D = 60, m E = 50, and m F = 70. 60 + 50 + 70 = 180 So, the sum of the angles of DEF is 180. 292 HotTopics
Finding the Measure of the EXAMPLE Unknown Angle in a Triangle S is a right angle, so its measure is 90. The measure of T is 35. Find the measure of U. 90 + 35 = 125 Add the two known angles. 180-125 = 55 Subtract the sum from 180. So, U = 55. The difference is the measure of the third angle. Find the measure of the third angle of each triangle. 15 40 110 16 45 17 60 60 Naming and Classifying Angles and Triangles 293
6 1 NAMING AND CLASSIFYING ANGLES AND TRIANGLES 6 1 Exercises Use the figure to answer Exercises 1 5. 1. Give six names for the line that passes through point P. 2. Name four rays that begin at point Q. 3. Name the right angle. 4. Find m PQT. 5. Find m PQS. Use the figure below to answer Exercises 6 8. 6. Identify a pair of complementary angles. 7. Identify a pair of supplementary angles. 8. Identify a pair of vertical angles. Use the figure to answer Exercises 9 and 10. 40 100 25 9. Is MNO an acute, an obtuse, or a right triangle? 10. Is MNO a scalene, an isosceles, or an equilateral triangle? 294 HotTopics