THE NATURE OF GEOMETRY Copyright © Cengage Learning. All rights reserved. 7

7.3 Triangles

3 Terminology

4 Every triangle has six parts: three sides and three angles. We name the sides by naming the endpoints of the line segments, and we name the angles by identifying the vertex (see Figure 7.30). A standard triangle showing the six parts Figure 7.30

5 Terminology Triangles are classified both by sides and by angles (single, double, and triple marks are used to indicate segments of equal length):

6 Terminology cont’d

7 Terminology We say that two triangles are congruent if they have the same size and shape. Suppose that we wish to construct a triangle with vertices D, E, and F, congruent to  ABC as shown in Figure A standard triangle showing the six parts Figure 7.30

8 Terminology We would proceed as follows (as shown in Figure 7.31): 1. Mark off segment DE so that it is congruent to AB. We write this as 2. Construct angle E so that it is congruent to angle B. We write this as Figure 7.31 Constructing congruent triangles

9 Terminology 3. Mark off segment You can now see that, if you connect points D and F with a straightedge, the resulting  DEF has the same size and shape as  ABC. The procedure we used here is called SAS, meaning we constructed two sides and an included angle (an angle between two sides) congruent to two sides and an included angle of another triangle. We call these corresponding parts.

10 Terminology There are other procedures for constructing congruent triangles; some of these are discussed in the problem set. For this example, we say  ABC  DEF. From this we conclude that all six corresponding parts are congruent.

11 Example 1 – Corresponding angles with congruent triangles Name the corresponding parts of the given triangles. a.  ABC  ABC b.  RST  UST Solution: a. corresponds to b. corresponds to corresponds to corresponds to

12 Example 1 – Solution corresponds to corresponds to cont’d

13 Angles of a Triangle

14 Angles of a Triangle One of the most basic properties of a triangle involves the sum of the measures of its angles.

15 Example 3 – Use algebra to find angles in a triangle Find the measures of the angles of a triangle if it is known that the measures are x, 2x – 15, and 3(x + 7) degrees. Solution: Using the theorem for the sum of the measures of angles in a triangle, we have x + (2x – 15) + 3(x + 17) = 180 x + 2x – x + 51 = 180 6x + 36 = 180 Sum of the measures of the angles is 180°. Eliminate parentheses. Combine similar terms.

16 Example 3 – Solution 6x = 144 x = 24 Now find the angle measures: x = 24 2x – 15 = 2(24) – 15 = 33 3(x + 17) = 3( ) = 123 The angles have measures of 24°, 33°, and 123°. Subtract 36 from both sides. cont’d Divide both sides by 6.

17 Angles of a Triangle An exterior angle of a triangle is the angle on the other side of an extension of one side of the triangle. An example is the angle whose measure is marked as x in Figure Figure 7.34 Exterior angle x

18 Angles of a Triangle Notice that the following relationships are true for any  ABC with exterior angle x: m  A + m  B + m  C = 180° and m  C + x = 180° Thus, m  A + m  B + m  C = m  C + x m  A + m  B = x Subtract m  C from both sides.

19 Angles of a Triangle

20 Example 4 – Find the exterior angle Find the value of x in Figure Solution: = 105 = x The measure of the exterior angle is 105°. Figure 7.35 What is x?

21 Isosceles Triangle Property

22 Isosceles Triangle Property In an isosceles triangle, there are two sides of equal length and the third side is called its base. The angle included by its legs is called the vertex angle, and the angles that include the base are called base angles. There is an important theorem in geometry that is known as the isosceles triangle property.

23 Isosceles Triangle Property In other words, if a triangle is isosceles, then the base angles have equal measures. The converse is also true; namely, if two angles of a triangle are congruent, the sides opposite them have equal length.

24 Example 5 – Equiangular implies equilateral Give a reasonable argument to prove that if a triangle is equiangular, it is also equilateral. Solution: If  ABC is equiangular, then m  A = m  B = m  C Since m  A = m  B from the converse of the isosceles triangle property, we have Again, since m  B = m  C, we have. Thus, we see that all three sides have the same length, and consequently  ABC is equilateral.