Name LESSON 4-2 Date Class Reteach Angle Relationships in Triangles According to the Triangle Sum Theorem, the sum of the angle measures of a triangle is 180°. * mJ mK mL 62 73 45 180° , The corollary below follows directly from the Triangle Sum Theorem. Corollary + Example The acute angles of a right triangle are complementary. mC 90 39 51° # % $ mC mE 90° Use the figure for Exercises 1 and 2. 1. Find mABC. ! 47° 2. Find mCAD. $ 38° " # Use RST for Exercises 3 and 4. 2 3. What is the value of x? X 14 X 4. What is the measure of each angle? mR 85°; mS 30°; mT 65° 4 X 3 What is the measure of each angle? 7 ! - , . 5. L " 5 # 6. C 49° Copyright © by Holt, Rinehart and Winston. All rights reserved. X 6 7. W 39.8° 14 (90 x)° Holt Geometry Name Date Class Reteach LESSON 4-2 Angle Relationships in Triangles An exterior angle of a triangle is formed by one side of the triangle and the extension of an adjacent side. continued remote interior angles exterior angle 1 and 2 are the remote interior angles of 4 because they are not adjacent to 4. Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. m4 m1 m2 Third Angles Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent. Find each angle measure. & $ X * # ' ( 8. mG X " ! 9. mD 51° 41° Find each angle measure. , . 5 0 X X 1 X 4 + X - 10. mM and mQ 2 11. mT and mR 82°; 82° Copyright © by Holt, Rinehart and Winston. All rights reserved. 3 33°; 33° 15 Holt Geometry Name Date LESSON 4-2 Class Name Practice A 4-2 Use the figure for Exercises 1–3. Name all the angles that fit the definition of each vocabulary word. 1. exterior angle 2. remote interior angles to �6 3. interior angle �1, �4, �6 �2, �3 �2, �3, �5 � � � For Exercises 4–7, fill in the blanks to complete each theorem or corollary. equiangular 4. The measure of each angle of an triangle is 60°. 180� 5. The sum of the angle measures of a triangle is right exterior angle 6. The acute angles of a . triangle are complementary. 7. The measure of an of the measures of its remote interior angles. 2. The acute angles of right triangle ABC are congruent. Find their measures. of a triangle is equal to the sum � 35° � � � 20° � 130° 60� Find each angle measure. 40° 12. m�P � (5� � 1)° � 35� � 9. m�E and m�G � 120� � 44�; 44� � � � � (9� � 9)° � 11. In �ABC and �DEF, m�A � m�D and m�B � m�E. Find m�F if an exterior angle at A measures 107�, m�B � (5x � 2)�, and m�C � (5x � 5)�. ��� ��� Date 108�; 108� (10� � 2)° (6� � 4)° � ���� 33�; 66�; 81� 10. m�T and m�V � (5� � 4)° 12. The angle measures of a triangle are in the ratio 3 : 4 : 3. Find the angle measures of the triangle. 110� 11 � 47� 7. m�PRS � 14. When a person’s joint is injured, the person often goes through rehabilitation under the supervision of a doctor or physical therapist to make sure the joint heals well. Rehabilitation involves stretching and exercises. The figure shows a leg bending at the knee during a rehabilitation session. Use what you know about triangles to find the angle measure that the knee is bent from the horizontal (fully extended) position. Copyright © by Holt, Rinehart and Winston. All rights reserved. 60� 8. In �LMN, the measure of an exterior angle at N measures 99�. m�L � _1_x � and m�M � _2_x�. Find m�L, m�M, and m�LNM. 3 3 � 13. m�VWY Name � � � 6. m�B 80° � 23° 65� 35° � 4-2 � � � 120° � 11. m�L � � Class Holt Geometry 12 Copyright © by Holt, Rinehart and Winston. All rights reserved. Name Practice C LESSON 4-2 Angle Relationships in Triangles Date Reteach Angle Relationships in Triangles � ��� � 180° � 2. Use this figure to write a flowchart proof that the sum of the measures of the exterior angles of a triangle, one at each vertex, is 360�. Corollary Reasons 1. Given 2. Construction 3. Triangle Sum Thm. m�C � 90 � 39 � 51° � m�C � m�E � 90° 5. Angle Add. Post. Use the figure for Exercises 1 and 2. 6. Subst. 1. Find m�ABC. � 47° 2 3 � � ��� � 4 1 ��� ��� Example The acute angles of a right triangle are complementary. 4. Add. Prop. of � ����������������� ����������������� ��������������� ����������� 6 ��� 2. Find m�CAD. 5 � 38° ��� ��� � � Use �RST for Exercises 3 and 4. 3. What is the value of x ? ������������������� ������������������ ��������������� ���������� ������ 3. Find the sum of the exterior angles, one at each vertex, of a quadrilateral. m�R � 85°; m�S � 30°; m�T � 65° ����� Copyright © by Holt, Rinehart and Winston. All rights reserved. � ��� � 5. �L �� �� �� ��������� � � � � interior � 540�; exterior � 360� ��� � What is the measure of each angle? 360� 4. Use the techniques you developed in Exercises 1–3 to find the sums of the measures of the interior angles and of the exterior angles, one at each vertex, of a pentagon. 13 ���������� ��������� 4. What is the measure of each angle? ���������������������� 5. A landscape artist plans to draw a pair of mountains. He wants his drawing to be reasonably accurate, so he takes some measurements and draws this figure. Find x, y, and z. � 14 ������������������� ������������������ ��������� ���������������������� Copyright © by Holt, Rinehart and Winston. All rights reserved. � The corollary below follows directly from the Triangle Sum Theorem. Statements 1. Quadrilateral ABCD 2. Draw AC. 3. m�D � m�DAC � m�DCA � 180°, m�B � m�BAC � m�BCA � 180� 4. m�D � m�DAC � m�DCA � m�B � m�BAC � m�BCA � 360� 5. m�DAC � m�BAC � m�DAB, m�DCA � m�BCA � m�DCB 6. m�D � m�DAB � m�B � m�DCB � 360� Holt Geometry Class m�J � m�K � m�L � 62 � 73 � 45 � 55� 54�; 72�; 54� According to the Triangle Sum Theorem, the sum of the angle measures of a triangle is 180°. 1. Write a two-column proof that the sum of the angle measures of a quadrilateral is 360�. Begin by drawing quadrilateral ABCD. (Hint: You will have to draw one � auxiliary line.) � Possible answer: 89.7� 5. 0.3� (9� � 2)° 65° � � LESSON z� 4. (90 � z )� � 10. m�G �° 45.1� 3. 44.9� 70� 9. m�F � � � � 115� 8. m�B 45� The measure of one of the acute angles in a right triangle is given. Find the measure of the other acute angle. Find the measure of each angle. 30° Angle Relationships in Triangles 1. An area in central North Carolina is known as the Research Triangle because of the relatively � Durham large number of high-tech companies and research 10.7 mi universities located there. Duke University, the 21.4 mi Chapel University of North Carolina at Chapel Hill, and Hill 25.7 mi North Carolina State University are all within this Raleigh area. The Research Triangle is roughly bounded by the cities of Chapel Hill, Durham, and Raleigh. From Chapel Hill, the angle between Durham and Raleigh measures 54.8�. From Raleigh, the angle between Chapel Hill and Durham measures 24.1�. Find the angle between 101.1� Chapel Hill and Raleigh from Durham. � � � Class Practice B LESSON Angle Relationships in Triangles � Date ��� Holt Geometry Copyright © by Holt, Rinehart and Winston. All rights reserved. 69 � � 6. �C 49° x � 93; y � 52; z � 35 � �� � 7. �W 39.8° 14 (90 � x)° Holt Geometry Holt Geometry Name Date Class Name Reteach LESSON 4-2 Date Challenge LESSON Angle Relationships in Triangles An exterior angle of a triangle is formed by one side of the triangle and the extension of an adjacent side. 4-2 continued remote interior angles Analyzing Verbal Descriptions and Using Auxiliary Figures Find the measure of each angle. exterior angle �1 and �2 are the remote interior angles of �4 because they are not adjacent to �4. � 88° 39° _ m�4 � m�1 � m�2 � � _ BA � DE. Explain how you could draw one or more auxiliary figures to help you find the value of x. Then find the value of x. Explain. � � 3. � � ��� � Third Angles Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent. � � � � _ � ��� ���� �� � � � � � 41° � � � ���������� ���������� � ��������� �� � � � 11. m�T and m�R 33°; 33° 82°; 82° 15 Copyright © by Holt, Rinehart and Winston. All rights reserved. Name LESSON 4-2 � � � 10. m�M and m�Q Date Class Holt Geometry Problem Solving 4. � lines � alt. int. � � 16 Holt Geometry Class Graphic Organizer �° 155° ��������������� ��������������������� ���������������������������� ���������������������������� ������������������������������� ����������������������������� � � � � �������������������� ��������������� � �������������������������������� ����������������������������� ���������������������� � � Use the figure of the banner for Exercises 3 and 4. ��������� ���������������������� 3. What is the value of n? � n � 12 4. What is the measure of each angle in the banner? � ���������������������� �������������������������� �������������������������������� ��������������������������� ��������������������������� ��������� � � � � � ��� � � �� � 30° � � 60° 60° 30° 2. Find m�TRP. 3. Find m�RTS. Use the figure for Exercises 4–7. 113° � The figure shows a path through a garden. Choose the best answer. � 1. Find m�QRP. �� 6. At takeoff, a ° � 23°. What is c °, the measure of the angle the pole makes with the athlete’s body? 32° � Use the given information to find the measures of the angles. �S and �Q are right angles. m�QPR � 30° �TRP is equiangular. �� � ����������������������� ������������� ��������������� Use the figure of the athlete pole vaulting for Exercises 5 and 6. ������������������� ��������������������������������� ���������������������������������� ��������������������������������������� ��������������� � ��������� 76°, 76°, 28° � 80° � 7. What is the measure of �QLP? A 20° C 110° B 70° D 125° � 55° 30° � � 82° 55° � 17 Copyright © by Holt, Rinehart and Winston. All rights reserved. (2� � 20)° (3� � 10)° � � 4. Find m�A. � 5. Find m�B. 9. What is the measure of �PMN? A 98° C 60° B 68° D 55° 8. What is the measure of �LPM? F 85° H 95° G 90° J 125° Copyright © by Holt, Rinehart and Winston. All rights reserved. Date This graphic organizer describes the relationships of interior and exterior angles in a triangle. 59° 5. What is x °, the measure of the angle that the pole makes when it first touches the ground? � � Reading Strategies 4-2 146° �� 1 5. � Add. Post. 6. Subst. Prop. of � Copyright © by Holt, Rinehart and Winston. All rights reserved. 2. A large triangular piece of plywood is to be painted to look like a mountain for the spring musical. The angles at the base of the plywood measure 76° and 45°. What is the measure of the top angle that represents the mountain peak? �� � 5. m�BCD � m�3 � m�4 6. m�BCD � m�1 � m�2 LESSON 121° � 2 4. m�2 � m�4 Name Angle Relationships in Triangles 1. The locations of three food stands on a fair’s midway are shown. What is the measure of the angle labeled x °? � � Additional Answer: Proofs will vary. Given: �ABC with exterior angle �BCD 4 3 1 Prove: m�BCD � m�1 � m�2 � � � Proof: Statements Reasons 1. �ABC with exterior angle �BCD 1. Given 2. Through _ C, draw line � parallel 2. Through a point outside a line, there is exactly one line parallel to AB. to the given line. 3. � lines � corr. � � 3. m�1 � m�3 ��� � Find each angle measure. � � ��� � � 2 9. m�D 51° � 4. It is possible to prove the Exterior Angle Theorem by drawing an auxiliary line. Use the figure at the right to show how this might be done. Write a complete proof on a separate sheet of paper. (Hint: Think about the angles formed when parallel lines are cut by a transversal.) � ��������� � ��� ��� � ��� Through C, draw line � || BA. So � is also || DE. Then apply the Alt. Int. � Thm. twice, followed by the � Add. Post. x° � 55° � 62° � 117°. Find each angle measure. � � ����������� ��������� ������� � ��� � � 8. m�G 2. In �WXY, the measure of �X is 4° less than twice the measure of �W. The measure of �Y is 24° less than 3 _1_ times m�W. Find m�Y. 2 1. In �FGH, the measure of �H is 14° less than the measure of �F. The measure of �G is 25° more than 2 _1_ times the 3 measure of �F. Find m�F. � Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. Class 6. Find m�BCF. 7. Find m�EFD. Holt Geometry Copyright © by Holt, Rinehart and Winston. All rights reserved. 70 � 40° 80° 120° 60° 18 Holt Geometry Holt Geometry
© Copyright 2024