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Name LESSON 4-2 Date Class Practice B 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. 2. The acute angles of right triangle ABC are congruent. Find their measures. 45 The measure of one of the acute angles in a right triangle is given. Find the measure of the other acute angle. 45.1 3. 44.9 4. (90 z) z 89.7 5. 0.3 Find each angle measure. 0 $ # 120° 23° (5X 1)° (9X 2)° ! 2 3 " 60 6. mB 1 47 7. mPRS 8. In LMN, the measure of an exterior angle at N measures 99. 1 x and mM __ 2 x. Find mL, mM, and mLNM. mL __ 3 3 9. mE and mG 44; 44 10. mT and mV 4 % 5 & (5X 4)° 108; 108 (10N 2)° (6X 4)° $ 33; 66; 81 7 ( (9N 9)° ' 6 11. In ABC and DEF, mA mD and mB mE. Find mF if an exterior angle at A measures 107, mB (5x 2), and mC (5x 5). 12. The angle measures of a triangle are in the ratio 3 : 4 : 3. Find the angle measures of the triangle. Copyright © by Holt, Rinehart and Winston. All rights reserved. 12 55 54; 72; 54 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