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Name LESSON 4-2 Date Class Practice A Angle Relationships in Triangles 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. 4. The measure of each angle of an equiangular triangle is 60°. 180⬚ 5. The sum of the angle measures of a triangle is 6. The acute angles of a right exterior angle . triangle are complementary. 7. The measure of an of the measures of its remote interior angles. of a triangle is equal to the sum Find the measure of each angle. ! 30° # 35° $ " ' 70⬚ 9. m⬔F ( , 130° ) 65° * 60⬚ 10. m⬔G 0 X° % & 115⬚ 8. m⬔B 20° 65⬚ 11. m⬔L 3 1 + 40° 35° 9 5 2 12. m⬔P 80° 4 6 35⬚ 7 8 13. m⬔VWY 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. 120⬚ 110⬚ Copyright © by Holt, Rinehart and Winston. All rights reserved. 11 Holt Geometry 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. m⬔B 1 47⬚ 7. m⬔PRS 8. In 䉭LMN, the measure of an exterior angle at N measures 99⬚. 1 x ⬚ and m⬔M ⫽ __ 2 x⬚. Find m⬔L, m⬔M, and m⬔LNM. m⬔L ⫽ __ 3 3 9. m⬔E and m⬔G 44⬚; 44⬚ 10. m⬔T and m⬔V 4 % 5 & (5X 4)° 108⬚; 108⬚ (10N 2)° (6X 4)° $ 33⬚; 66⬚; 81⬚ 7 ( (9N 9)° ' 6 11. In 䉭ABC and 䉭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)⬚. 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 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°. * m⬔J ⫹ m⬔K ⫹ m⬔L ⫽ 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. m⬔C ⫽ 90 ⫺ 39 ⫽ 51° # % $ m⬔C ⫹ m⬔E ⫽ 90° Use the figure for Exercises 1 and 2. 1. Find m⬔ABC. ! 47° 2. Find m⬔CAD. $ 38° " # Use 䉭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? m⬔R ⫽ 85°; m⬔S ⫽ 30°; m⬔T ⫽ 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. m⬔4 ⫽ m⬔1 ⫹ m⬔2 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. m⬔G X " ! 9. m⬔D 51° 41° Find each angle measure. , . 5 0 X X 1 X 4 + X - 10. m⬔M and m⬔Q 2 11. m⬔T and m⬔R 82°; 82° Copyright © by Holt, Rinehart and Winston. All rights reserved. 3 33°; 33° 15 Holt Geometry