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Name LESSON 3-3 Date Class Practice B Proving Lines Parallel Use the figure for Exercises 1–8. Tell whether lines m and n must be parallel from the given information. If they are, state your reasoning. (Hint: The angle measures may change for each exercise, and the figure is for reference only.) 1. 7 3 M N 1 8 2 7 3 6 4 5 2. m3 (15x 22)°, m1 (19x 10), x8 m 储 n; Conv. of Alt. Int. ⭄ Thm. m 储 n; Conv. of Corr. ⭄ Post. 3. 7 6 4. m2 (5x 3)°, m3 (8x 5), x 14 m and n are parallel if and only if m 储 n; Conv. of Same-Side m⬔7 90°. Int. ⭄ Thm. 5. m8 (6x 1)°, m4 (5x 3)°, x 9 6. 5 7 m 储 n; Conv. of Corr. ⭄ Post. m and n are not parallel. 7. 1 5 8. m6 (x 10)°, m2 (x 15) m 储 n; Conv. of Alt. Ext. ⭄ Thm. m and n are not parallel. 9. Look at some of the printed letters in a textbook. The small horizontal and vertical segments attached to the ends of the letters are called serifs. Most of the letters in a textbook are in a serif typeface. The letters on this page do not have serifs, so these letters are in a sans-serif typeface. (Sans means “without” in French.) The figure shows a capital letter A with serifs. Use the given information to write a _ _ paragraph proof that the serif, segment HI , is parallel to segment JK . Given: 1 and 3 are supplementary. _ _ Prove: HI JK * ( 1 2 3 + ) Sample answer: The given information states that ⬔1 and ⬔3 are supplementary. ⬔1 and ⬔2 are also supplementary by the Linear Pair Theorem. Therefore ⬔3 and ⬔2 must be congruent by the_ Congruent _ Supplements Theorem. Since ⬔3 and ⬔2 are congruent, HI and JK are parallel by the Converse of the Corresponding Angles Postulate. Copyright © by Holt, Rinehart and Winston. All rights reserved. 20 Holt Geometry Name Date LESSON 3-3 Class Name Practice A LESSON 3-3 Proving Lines Parallel 1. The Converse of the Corresponding Angles Postulate states that if two coplanar lines are cut by a transversal so that a pair of corresponding angles is congruent, parallel . then the two lines are Use the figure for Exercises 2 and 3. Given the information in each exercise, state the reason why lines b and c are parallel. 2. �4 � �8 1 2 3 4 5 6 7 8 � � Practice B Proving Lines Parallel 1. �7 � �3 4. m�2 � (5x � 3)°, m�3 � (8x � 5)�, x � 14 Corr. � Post. so that a pair of congruent alternate interior angles are supplementary , then the two lines are parallel. parallel angles are congruent, then the two lines are � 3 2 Given: �1 and �3 are supplementary. Prove: m � n � 1 Proof: . �2 and �3 are supplementary. _ � � 1 2 3 � � Sample answer: The given information states that �1 and �3 are supplementary. �1 and �2 are also supplementary by the Linear Pair Congruent Theorem. Therefore �3 and �2 must be congruent by the_ _ Supplements Theorem. Since �3 and �2 are congruent, HI and JK are parallel by the Converse of the Corresponding Angles Postulate. 2. Linear Pair Thm. m�n _ Prove: HI � JK Given � Supps. Thm. 3. c. m and n are not parallel. Given: �1 and �3 are supplementary. Reasons 1. a. 3. �1 � �2 4. Conv. of Corr. � Post. 19 Copyright © by Holt, Rinehart and Winston. All rights reserved. Name 3-3 m � n; Conv. of Corr. � Post. 8. m�6 � (x � 10)°, m�2 � (x � 15)� m � n; Conv. of Alt. Ext. � Thm. m�n �2 and �3 are supplementary. Given � Supps. Thm. Statements 1. �1 and �3 are supplementary. LESSON 6. �5 � �7 9. Look at some of the printed letters in a textbook. The small horizontal and vertical segments attached to the ends of the letters are called serifs. Most of the letters in a textbook are in a serif typeface. The letters on this page do not have serifs, so these letters are in a sans-serif typeface. (Sans means “without” in French.) The figure shows a capital letter A with serifs. Use the given information to write a _ _ paragraph proof that the serif, segment HI, is parallel to segment JK. 7. Shu believes that a theorem is missing from the lesson. His conjecture is that if two coplanar lines are cut by a transversal so that a pair of same-side exterior angles are supplementary, then the two lines are parallel. Complete the two-column proof with the statements and reasons provided. 4. d. Int. � Thm. 7. �1 � �5 6. If two coplanar lines are cut by a transversal so that a pair of alternate exterior 2. b. m � n; Conv. of Same-Side m�7 � 90°. m and n are not parallel. 5. If two coplanar lines are cut by a transversal so that a pair of same-side interior angles are m and n are parallel if and only if 5. m�8 � (6x � 1)°, m�4 � (5x � 3)°, x � 9 , then the two lines are parallel. 1 8 2 7 3 6 4 5 m � n; Conv. of Corr. � Post. 3. �7 � �6 Fill in the blanks to complete these theorems about parallel lines. transversal � � 2. m�3 � (15x � 22)°, m�1 � (19x � 10)�, x�8 m � n; Conv. of Alt. Int. � Thm. m�7 = 68°, �3 � �7, Conv. of 4. If two coplanar lines are cut by a Class Use the figure for Exercises 1–8. Tell whether lines m and n must be parallel from the given information. If they are, state your reasoning. (Hint: The angle measures may change for each exercise, and the figure is for reference only.) 3. m�3 � 68�, m�7 � (5x + 3)�, x � 13 Conv. of Corr. � Post. Date Date Holt Geometry Class Name Practice C LESSON 3-3 Proving Lines Parallel 1. p � q, m�1 � (6x � y � 4)�, m�2 � (x � 9y � 1)�, m�3 � (11x � 2)� Find x, y, and the measures of �1, �2, and �3. 1 � 20 Copyright © by Holt, Rinehart and Winston. All rights reserved. 2. Use the figure and the given information to write a paragraph proof that the sum of the measures of the three angles in a triangle is _ 180�. ‹__› (Hint: Begin by constructing FG through point C and parallel to AB.) � 1 You can use the Converse of the Corresponding Angles Postulate to show that two lines are parallel. � Given: ABC is a triangle. 4 � Prove: m�1 � m�2 � m�3 � 180� ‹__› 2 �1 � �3 � q || r Possible answer: Construct FG through point C and parallel to AB . �3 and �4 are a linear pair, so m�3 � m�4 � 180� by the Linear Pair Theorem. But the Angle Addition Postulate shows that m�4 � m�ACF � m�FCD, so by substitution m�3 � m�ACF � m�FCD � 180�. m�1 � m�ACF by the Alternate Interior Angles Theorem and m�2 � m�FCD by the Corresponding Angles Postulate. Therefore m�1 � m�2 � m�3 � 180� by substitution. 3. In an isosceles triangle, at least two of the‹__ angles are __› congruent. To construct › �FDE and let isosceles triangle DEH, begin by drawing DE and DF. If you copy __ › the angle open in the same be› parallel to DF . Instead, ___› direction, the ray would __ intersects copy �FDE and draw EG so that the ray_ _DF. Label the intersection point H. Use your compass to measure DH and EH. What is remarkable about the measures of these segments? The measures of the segments are equal. 1 2 � 34 � �1 � �3 are corresponding angles. Converse of the Corresponding Angles Postulate Given: m�2 � 3x°, m�4 � (x � 50)°, x � 25 _ m�2 � 3(25)° � 75° Substitute 25 for x. m�4 � (25 � 50)° � 75° Substitute 25 for x. m�2 � m�4 Transitive Property of Equality �2 � �4 Definition of congruent angles q || r Converse of the Corresponding Angles Postulate For Exercises 1 and 2, use the Converse of the Corresponding Angles Postulate and the given information to show that c || d. 1. Given: �2 � �4 �2 � �4 c || d � � If two coplanar lines are cut by a transversal so that a pair of corresponding angles are congruent, then the two lines are parallel. Given: �1 � �3 � 3 � � Holt Geometry Proving Lines Parallel � x � 11; y � �5; m�1 � 57°; m�2 � 57°; m�3 � 123� Class Reteach Converse of the Corresponding Angles Postulate 2 3 Date �2 and �4 are corr. �. Conv. of Corr. � Post. 1 3 2 4 � � � 2. Given: m�1 � 2x°, m�3 � (3x � 31)°, x � 31 � m�1 � 2x° � 2(31)° � 62° m�3 � (3x � 31)° � 3(31)° � 31° � 62° m�1 � m�3 �1 � �3 c || d � 4. Construct another isosceles triangle with angles and side lengths different from the triangle you drew in Exercise 3. Again measure the lengths of the sides opposite the congruent angles. Write a conjecture about the measures of the side lengths in isosceles triangles. Possible answer: If a triangle is isosceles, then the sides opposite the �������� ������� congruent angles are congruent. Copyright © by Holt, Rinehart and Winston. All rights reserved. 21 Copyright © by Holt, Rinehart and Winston. All rights reserved. Holt Geometry Copyright © by Holt, Rinehart and Winston. All rights reserved. 55 Substitute 31 for x. Substitute 31 for x. Trans. Prop. of � Def. of � � Conv. of Corr. � Post. 22 Holt Geometry Holt Geometry