Angle Theorems for Triangles 8.8.D

Transcription

1 ? LSSON 7.2 ngle Theorems for Triangles SSNTIL QUSTION xpressions, equations, and relationships 8.8. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, What can you conclude about the measures of the angles of a triangle? XPLOR TIVITY 8.8. Sum of the ngle Measures in a Triangle There is a special relationship between the measures of the interior angles of a triangle. raw a triangle and cut it out. Label the angles,, and. Tear off each corner of the triangle. ach corner includes the vertex of one angle of the triangle. rrange the vertices of the triangle around a point so that none of your corners overlap and there are no gaps between them. What do you notice about how the angles fit together around a point? F What is the measure of a straight angle? escribe the relationship among the measures of the angles of. Houghton Mifflin Harcourt Publishing ompany The Triangle Sum Theorem states that for, m + m + m =. Reflect. Justify Reasoning an a triangle have two right angles? xplain. 2. nalyze Relationships escribe the relationship between the two acute angles in a right triangle. xplain your reasoning. Lesson

2 XPLOR TIVITY Justifying the Triangle Sum Theorem You can use your knowledge of parallel lines intersected by a transversal to informally justify the Triangle Sum Theorem. Follow the steps to informally prove the Triangle Sum Theorem. You should draw each step on your own paper. The figures below are provided for you to check your work. raw a triangle and label the angles as, 2, and 3 as shown. raw line a through the base of the triangle. b The Parallel Postulate states that through a point not on a line l, there is exactly one line parallel to line l. raw line b parallel to line a, through the vertex opposite the base of the triangle. a 2 3 xtend each of the non-base sides of the triangle to form transversal s and transversal t. Transversals s and t intersect parallel lines a and b. Label the angles formed by line b and the transversals as 4 and 5. b s t 4 5 F ecause 4 and are alternate interior a 2 3 angles, they are. Label 4 with the number of the angle to which it is congruent. G ecause 5 and they are. are alternate interior angles, H Label 5 with the number of the angle to which it is congruent. The three angles that lie along line b at the vertex of the triangle are, 4, and 5. Notice that these three angles lie along a line. So, m + m 4 + m 5 =. ecause angles 2 and 4 are congruent and angles 3 and 5 are congruent, you can substitute m 2 for m 4 and m 3 for m 5 in the equation above. So, m + m 2 + m 3 =. This shows that the sum of the angle measures in a triangle is always. Houghton Mifflin Harcourt Publishing ompany 200 Unit 3

3 Reflect 3. nalyze Relationships How can you use the fact that m 4 + m + m 5 = 80 to show that m 2 + m + m 3 = 80? Finding Missing ngle Measures in Triangles If you know the measures of two angles in a triangle, you can use the Triangle Sum Theorem to find the measure of the third angle. XMPL Find the missing angle measure Math On the Spot My Notes STP Write the Triangle Sum Theorem for this triangle. 00 m + m + m F = 80 STP 2 Substitute the given angle measures m + 55 = 80 F 55 STP 3 Solve the equation for m m + 55 = 80 Houghton Mifflin Harcourt Publishing ompany YOUR TURN 55 + m = m = 25 So, m = 25. Find the missing angle measure. 4. K 5. Subtract 55 from both sides. Simplify. S 29 R J Personal 7 56 L 6 Math Trainer m K = m R = T Online ssessment and Intervention Lesson

4 XPLOR TIVITY xterior ngles and Remote Interior ngles n interior angle of a triangle is formed by two sides of the triangle. n exterior angle is formed by one side of the triangle and the extension of an adjacent side. ach exterior angle has two remote interior angles. remote interior angle is an interior angle that is not adjacent to the exterior angle. 2, 2, and 3 are interior angles. 4 is an exterior angle. 3 4 and 2 are remote interior angles to 4. There is a special relationship between the measure of an exterior angle and the measures of its remote interior angles. 2 xtend the base of the triangle and label the exterior angle as 4. The Triangle Sum Theorem states: 3 m + m 2 + m 3 =. 3 and 4 form a, so m 3 + m 4 =. Use the equations in and to complete the following equation: m + m 2 + = + m 4 Use properties of equality to simplify the equation in : The xterior ngle Theorem states that the measure of an is equal to the sum of the measure of its Reflect 6. Sketch a triangle and draw all of its exterior angles. How many exterior angles does a triangle have at each vertex? angles. angle Houghton Mifflin Harcourt Publishing ompany 7. How many total exterior angles does a triangle have? 202 Unit 3

5 Using the xterior ngle Theorem You can use the xterior ngle Theorem to find the measures of the interior angles of a triangle. XMPL Math On the Spot Find m and m. STP Write the xterior ngle Theorem as it applies to this triangle. m + m = m (4y - 4) 52 3y STP 2 STP 3 Substitute the given angle measures. (4y - 4) + 3y = 52 Solve the equation for y. (4y - 4) + 3y = 52 Math Talk Mathematical Processes escribe two ways to find m. 4y y = 52 7y - 4 = 52 _ + 4 _ + 4 7y = 56 7y 7 = 56 7 y = 8 Remove parentheses. Simplify. dd 4 to both sides. Simplify. ivide both sides by 7. Simplify. STP 4 Use the value of y to find m and m. Houghton Mifflin Harcourt Publishing ompany YOUR TURN m = 4y - 4 m = 3y = 4(8) - 4 = 3(8) = 32-4 = 24 = 28 So, m = 28 and m = Find m M and m N. m M = m N = M (5y + 3) (4y + 8) 46 N P Q Personal Math Trainer Online ssessment and Intervention Lesson

6 Guided Practice Find each missing angle measure. (xplore ctivity and xample ). M 2. R 26 L 78 3 N Q 24 S m M = m Q = Use the Triangle Sum Theorem to find the measure of each angle in degrees. (xplore ctivity 2 and xample ) 3. T (7x + 4) (5x + 3) V 4. X n Y n 2 U (2x + 5) m T =, m U =, m V = Z n 2 m X =, m Y =, m Z = Use the xterior ngle Theorem to find the measure of each angle in degrees. (xplore ctivity 3 and xample 2) 5. (7y + 6) 6. J K L 6 (8z + 3)? 4y 6 F (5z - 3) M m =, m =, m L =, m M =, m = m LKM = SSNTIL QUSTION HK-IN 7. escribe the relationships among the measures of the angles of a triangle. Houghton Mifflin Harcourt Publishing ompany 204 Unit 3

7 Name lass ate 7.2 Independent Practice 8.8. Personal Math Trainer Online ssessment and Intervention Find the measure of each angle T 98 2x x x F x W m = m T = V m F = m V = 0. G. Q 5x (3y + 5) S 53 R (2y - 7) P H 4x 3x J m Q = m G = m H = m P = m QRP = m J = Houghton Mifflin Harcourt Publishing ompany m = L 3x K 2x x m K = m L = M N m = m KML = m = m LMN = 4. Multistep The second angle in a triangle is five times as large as the first. The third angle is two-thirds as large as the first. Find the angle measures. Lesson

8 5. nalyze Relationships an a triangle have two obtuse angles? xplain. FOUS ON HIGHR ORR THINKING Work rea 6. ritical Thinking xplain how you can use the Triangle Sum Theorem to find the measures of the angles of an equilateral triangle. 7. a. raw onclusions Find the sum of the measures of the angles in quadrilateral. (Hint: raw diagonal _. How can you use the figures you have formed to find the sum?) Sum = b. Make a onjecture Write a Quadrilateral Sum Theorem. xplain why you think it is true. 8. ommunicate Mathematical Ideas escribe two ways that an exterior angle of a triangle is related to one or more of the interior angles. Houghton Mifflin Harcourt Publishing ompany 206 Unit 3