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Chapter 14 Lesson The Triangle-Sum Property 14-7 Vocabulary revolution interior angles of a polygon exterior angles of a polygon BIG IDEA In all triangles, the sum of the measures of the three angles is the same number. Ancient Babylonians realized that each night at the same time, stars are in a slightly different position than the night before. They have rotated a small amount. In a year, they rotate all the way to their original position, 1 because Earth goes around the sun in a year. So, they rotate _ of the 365 way around a circle each day. The number 365 is not an easy number to use in making calculations. Because 360 is divisible by 2, 3, 4, 5, 6, 8, 9, 10, and 12, it is an easier number to use. In fact, 360 is the smallest number divisible by all these numbers. So the Babylonians thought of 1 the stars as moving _ of the way around the sky each day. From this 360 we get that one full revolution is a rotation of magnitude 360º. Activity MATERIALS thin paper such as tracing paper, a pencil, and a ruler Step 1 Draw any triangle ABC and trace it onto the thin paper. Label the angles 1, 2, and 3 at points A, B, and C, as shown. A 1 B 3 2 C Step 2 Measure to find the midpoint M of BC. Rotate ABC 180º about M. (Move the triangle you originally drew and trace it again.) The preimage and image are shown. A' is the image of A. When you perform this rotation, where are the images of B and C? A SMP10_A_SE_C14_L07A_T_002 1 B 2 3 M 3 2 C 1 A' 36 Some Important Geometry Ideas SMP10_A_SE_C14_L07A_T_003 Lesson 14-7 Step 3 Measure to find the midpoint N of A'C. Rotate A'BC 180º about N. B' is the image of B. A 1 2 3 B M 3 C 2 1 N 3 1 2 B' A' Step 4 Measure to find the midpoint O of B'C. Rotate A'B'C 180º about O. A* is the image of A'. SMP10_A_SE_C14_L07A_T_004 A 1 2 3 B 3 C 2 13 M N O 3 1 2 A* 2 1 B' A' Step 5 Measure to find the midpoint P of A*C. Rotate A*B'C 180º about P. B* is the image of B'. B* A 1 1 2 3 B C 3 2 2 13 M N 2 A* O 3 1 2 3 P 1 B' A' Step 6 Measure to find the midpoint Q of B*C. Rotate A*B*C 180º about Q. Where is the image of A*? B* A 2 1 SMP10_A_SE_C14_L07A_T_006 3 Q 1 B 2 3 P C3 1 2 M 2 1 N 1 2 3 3 2 A* O 3 1 B' A' You should find that the triangles completely cover the region around point C. Is this result the same for your classmates who started with different triangles? SMP10_A_SE_C14_L07A_T_007 The Triangle-Sum Property 37 Chapter 14 In the Activity, notice that at point C, ∠1, ∠2, and ∠3 each appear twice. Together, these six angles form a full revolution around point C. One full revolution measures 360º. The Activity illustrates that 2(m∠1 + m∠2 + m∠3) = 360º, so m∠1 + m∠2 + m∠3 = 180º. This is a property of any triangle. Triangle-Sum Property The sum of the measures of the three angles of any triangle is 180º. QY QY What is the sum of the measures of the angles of an isosceles triangle? Example 1 Suppose two angles of a triangle measure 36.4º and 64.6º. What is the measure of the third angle of the triangle? Solution Let x be the measure of the third angle. The sum of the measures of the three angles is 180º, so xº + 36.4º + 64.6º = 180º. x + 101 = 180 x = 79 The third angle measures 79º. Applying the Triangle-Sum Property Recall that complementary angles are two angles whose measures add to 90º. In the diagram of Example 2, ∠TOP and ∠TOS are complementary angles, because m∠POS = 90º. P Example 2 In the diagram at the right, find x. 19º Solution x = m∠OTS, and ∠OTS is in OTS. So x + y + m∠TOS = 180°. m∠TOS = 43°. To find y, use POS. m∠P + m∠POS + y = 180° 19° + 90° + y = 180° T y = 71° x So x + 71° + 43° = 180° x = 66° 38 O 43º y S Some Important Geometry Ideas SMP08TM2_SE_C06_T_0287 Lesson 14-7 Angles formed by the sides of a polygon are its interior angles. Angles that form linear pairs with interior angles of a polygon are exterior angles of the polygon. In the diagram in Example 3, ∠ ABC, ∠BCA, and ∠CAB are interior angles. ∠DAC, ∠CBE, and ∠ACF are exterior angles. Example 3 Find the measure of ∠DAC in the drawing at the right. Solution Let x be the measure of ∠DAC. Let y be the measure of ∠BAC. Use the Triangle-Sum Property to find y. D A xº B yº E 23º 120º F C 23° + 120° + y° = 180° 143 + y = 180 y = 37 Arithmetic Add –143 to both sides. SMP08TM2_SE_C06_T_0136 Now use the fact the ∠DAC and ∠BAC form a linear pair. x° + y° = 180° x + 37 = 180 x = 143 Substitution Add –37 to both sides. So, m∠DAC = 143°. Notice ∠DAC = m∠ABC + ∠BCA. This is an example of the Exterior Angle Theorem for Triangles. Exterior Angle Theorem for Triangles In a triangle, the measure of an exterior angle is equal to the sum of the measures of the interior angles at the other two vertices of the triangle. The Triangle-Sum Property 39 Chapter 14 GUIDED Example 4 Use the information given in the drawing. Explain the steps needed to find m∠9. 105˚ 2 1 3 5 7 4 6 8 9 Solution The drawing is complicated. Examine it carefully. You may wish to make an identical drawing and write in angle measures as they are found. Here is one way to solve the problem. The arrows indicate that two of the lines are parallel. ∠5 and the angle that measures 105° are corresponding angles. So m∠5 = ? . ∠7 forms a linear pair with ∠5, so they are SMP08TM2_SE_C06_T_0137 supplementary. Thus m∠7 = ? . The triangle has a 90° angle created by the perpendicular lines, but it also has ∠7, a ? angle. The Triangle-Sum Property says that the measures of the three angles add to 180°. Therefore, m∠8 = ? . Finally, ∠9 and ∠8 form a linear pair, so m∠9 = ? . Questions COVERING THE IDEAS A 1 1. At the right is the result of Step 2 in the Activity. Why do the angles of A'BC have the same measures as the angles of ABC? 2. Multiple Choice Which is true? B A In some but not all triangles, the sum of the measures of the angles is 180º. B In all triangles, the sum of the measures of the angles is 180º. 2 3 M 3 2 C 1 A' C The sum of the measures of the angles of a triangle can be any number from 180º to 360º. D If two angles of a triangle are complementary, then the measure of SMP10_A_SE_C14_L07A_T_003 the third angle must be greater than 90º. In 3–6, two angles of a triangle have the given measures. Find the measure of the third angle. 3. 45º, 45º 4. 2º, 3º 5. 70º, 80º 40 Some Important Geometry Ideas 6. xº, 120º - xº Lesson 14-7 7. In Example 3, find a. m∠ ACF. b. m∠FBE. c. m∠ ACF + m∠FBE + m∠CAD. 8. In Example 4, explain how to find m∠4. APPLYING THE MATHEMATICS a. If m∠1 = 40º and m∠2 = 60º, fill in the measures of all the angles in the six triangles. b. Give 5 pairs of parallel lines. c. How do you know the lines are parallel? 1 G B 2 outlined in the kaleidoscope quilt block at the right, the vertex angle equals 54º. What is the measure of each base angle? 11. Find m∠DAB below. 12. Find m∠VYZ below. B E 3 D C 10. Many quilt patterns use triangles. In the isosceles triangle SMP10_A_SE_C14_L07A_T_008 W 54º 50º 50º 43º D F A 9. Copy the figure at the right. A V C Y Z 13. Three lines intersect forming ∠1, ∠2, ∠3, and ∠4 as shown below. Explain why m∠1 + m∠2 + m∠3 = 180º. SMP08TM2_SE_C06_T_0141 SMP08TM2_SE_C06_T_0140 3 4 1 2 8 14. Use the information given in the drawing at the right. Find the measures of angles 1 through 8. 3 15. Explain why a triangleSMP08TM2_SE_C06_T_0139 cannot have two right angles. 16. In an equilateral triangle, all the angles have the same measure. What is this measure? 123º 2 1 61º 4 5 7 6 SMP08TM2_SE_C06_T_0142 The Triangle-Sum Property 41 Chapter 14 −−− −−− 17. In the figure below, BA ⊥ AC. Angle BAC is bisected (split into , and m∠B = 70º. Find the measures of ∠1, two equal parts) by AD ∠2, and ∠3. B D 70º A 1 2 45º 45º 3 C 18. Multiple Choice Two angles of a triangle have measures a and b degrees. The third angle must have what measure? R P A 180 - a + b degrees B 180 + a + b degrees SMP08TM2_SE_C06_T_0143 58º C 180 + a - b degrees D 180 - a - b degrees −−− −− 19. In the figure at the right, PQ RS = T. Find the measures of as many angles in the figure as you can. 20º 20. Explain why the measure of an exterior angle must be equal to the sum of the measures of the interior angles at the other two vertices of the triangle. S 55º T 22º Q 21. a. Write a theorem about the sum of the measures of the exterior angles of a triangle. b. Explain why your theorem is true. SMP08TM2_SE_C06_T_0144 EXPLORATION 22. You have seen that the sum of the measures of angles in a triangle on a plane is 180º, but what about a triangle drawn along the surface of a globe or other sphere? Suppose you draw a triangle by starting at the North Pole and drawing a line south. Then draw a line west, then another line north back to the pole. Will the sum of the angles of a triangle on a sphere equal 180º, be greater than 180º, or be less than 180º? Experiment using a globe or other sphere and summarize your results. QY ANSWER 180° 42 Some Important Geometry Ideas