In Exercises 4 7, use the diagram to name each of the following. true

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1 Name lass ate - Reteaching Lines and Angles Not all lines and planes intersect. Planes that do not intersect are parallel planes. Lines that are in the same plane and do not intersect are parallel. The symol 6 shows that lines or planes are parallel: * A ) 6 * ) means Line A is parallel to line. Lines that are not in the same plane and do not intersect are skew. Parallel planes: plane A 6 plane E A plane 6 plane AE plane 6 plane AE Examples of parallel lines: * ) 6 * A ) 6 * E ) 6 * ) Examples of skew lines: * ) is skew to * ), * AE ), * E ), and * ). E In, use the figure at the right.. Shade one set of parallel planes. Answers may vary. Sample:. Trace one set of parallel lines with a solid line.. Trace one set of skew lines with a dashed line. In 4 7, use the diagram to name each of the following. * ) 4. a line that is parallel to RS * ) * ) Answers may vary. Sample: NO 5. a line that is skew to QU * ) Answers may vary. Sample: RS 6. a plane that is parallel to NRTP plane OSUQ * ) 7. three lines that are parallel to OQ * ) * ) * ) NP, RT, and SU In 8, descrie the statement as true or false. If false, explain. * ) * ) 8. plane IKJ 6 plane IEK 9. 6 K alse; these planes intersect. true * ) * ) * ) * ) J and are skew lines.. 6 KI 0. alse; the lines intersect. J alse; the lines are in different planes, and since they do not intersect they are skew. P T R N Q U K E I O S opyright y Pearson Education, Inc., or its affiliates. All Rights Reserved. 9

2 Name lass ate - Reteaching (continued) Lines and Angles The diagram shows lines a and intersected y line x. Line x is a transversal. A transversal is a line that intersects two or more lines found in the same plane. The angles formed are either interior angles or exterior angles. a x Interior Angles etween the lines cut y the transversal /, /4, /5, and /6 in diagram aove our types of special angle pairs are also formed. Exterior Angles outside the lines cut y the transversal /, /, /7, and /8 in diagram aove Angle Pair efinition Examples alternate interior alternate exterior same-side interior inside angles on opposite sides of the transversal, not a linear pair outside angles on opposite sides of the transversal, not a linear pair inside angles on the same side of the transversal and 6 4 and 5 and 8 and 7 and 5 4 and 6 corresponding in matching positions aove or elow the transversal, ut on the same side of the transversal and 5 and 7 and 6 4 and 8 Use the diagram at the right to answer 5.. Name all pairs of corresponding angles. l and l5; l and l6; l and l8; l4 and l7. Name all pairs of alternate interior angles. l4 and l6; l and l5 4. Name all pairs of same-side interior angles. l4 and l5; l and l6 5. Name all pairs of alternate exterior angles. l and l8; l and l7 t 4 s c Use the diagram at the right to answer 6 and 7. ecide whether the angles are alternate interior angles, same-side interior angles, corresponding, or alternate exterior angles. 6. / and /5 7. /4 and /6 alternate exterior angles same-side interior angles opyright y Pearson Education, Inc., or its affiliates. All Rights Reserved. 0

3 Name lass ate - Reteaching Properties of Parallel Lines When a transversal intersects parallel lines, special supplementary and congruent angle pairs are formed. Supplementary angles formed y a transversal intersecting parallel lines: same-side interior angles (Postulate -) m/4 m/ m/ m/ ongruent angles formed y a transversal intersecting parallel lines: alternate interior angles (Theorem -) /4 > /6 / > /5 corresponding angles (Theorem -) / > /5 / > /6 /4 > /7 / > / , m q alternate exterior angles (Theorem -) / > /8 / > /7 Identify all the numered angles congruent to the given angle. Explain.. l6; vert. '.. 4 c are O; l; corresp. ' are 5 d O; l4; alt. 7 ext. ' are O f a l; vert. ' are O; l5; x alt. int. ' are O; l7; corresp. ' are O. 4. Supply the missing reasons in the two-column proof. i iven: g 6 h, i 6 j Prove: / is supplementary to /6. j 4 i j l; vert. ' are O; l4; alt. ext. ' are O. e Statements Reasons ) / > / ) 9 Vertical angles are congruent. ) g 6 h; i 6 j ) iven ) / > / ) 9 orresponding angles are congruent. 4) / and /6 are supplementary. 4) 9 Same-side interior angles are supplementary. 5) / and /6 are supplementary. 5) 9 Sustitution property g h opyright y Pearson Education, Inc., or its affiliates. All Rights Reserved. 9

4 Name lass ate - Reteaching (continued) Properties of Parallel Lines You can use the special angle pairs formed y parallel lines and a transversal to find missing angle measures. If m/ 5 00, what are the measures of / through /8? Supplementary angles: m/ m/ 5 80 m/ m/ Vertical angles: m/ 5 m/ m/ 5 00 Alternate exterior angles: m/ 5 m/7 m/ Alternate interior angles: m/ 5 m/5 m/ orresponding angles: m/ 5 m/6 m/ Same-side interior angles: m/ m/ m/ What are the measures of the angles in the figure? (x 0) (x 5) 5 80 Same-Side Interior Angles Postulate 5x omine like terms. 5x 5 75 Sutract 5 from each side. (x 0) x 5 5 ivide each side y 5. ind the measure of these angles y sustitution. x 0 5 (5) x 5 5 (5) x 0 5 (5) To find m/, use the Same-Side Interior Angles Postulate: 50 m/ 5 80, so m/ 5 0 (x 0) (x 5) ind the value of x. Then find the measure of each laeled angle x (4x 0) (x 0) x 50; 50; 90 (x 5) 5; 0; 50 (x 0) (8x 0) 40; 00; 80; 65 opyright y Pearson Education, Inc., or its affiliates. All Rights Reserved. 0

5 Name lass ate - Reteaching Proving Lines Parallel Special angle pairs result when a set of parallel lines is intersected y a transversal. The converses of the theorems and postulates in Lesson - can e used to prove that lines are parallel. Theorem -4: onverse of orresponding Angles Theorem If / > /5, then a a Theorem -5: onverse of the Alternate Interior Angles Theorem If / > /6, then a 6. Theorem -6: onverse of the Same-Side Interior Angles Postulate If / is supplementary to /5, then a 6. Theorem -7: onverse of the Alternate Exterior Angles Theorem If / > /7, then a 6. or what value of x is 6 c? The given angles are alternate exterior angles. If they are congruent, then 6 c. (x ) x 5 8 x 5 40 x 5 70 Which lines or line segments are parallel? Justify your answers. OP n QN ecause the O angles are alt. int. '... W X. N A Y Z Q O A n ecause the O' WX n YZ ecause the O are alt. ext. angles. ' are alt. int. '. ind the value of x for which g n h. Then find the measure of each laeled angle g g (6x) (x 6) (0x ) (x 6) h h 55; 6 7; 4; 8 8 P g c h (4x 8) 4; 78; 78 opyright y Pearson Education, Inc., or its affiliates. All Rights Reserved. 9

6 Name lass ate - Reteaching (continued) Proving Lines Parallel A flow proof is a way of writing a proof and a type of graphic organizer. Statements appear in oxes with the reasons written elow. Arrows show the logical connection etween the statements. Write a flow proof for Theorem -: If a transversal intersects two parallel lines, then alternate interior angles are congruent. iven: / 6 m Prove: / > / t, m, m iven If lines, then corresponding are. Vertical angles are. Transitive Property of omplete a flow proof for each. 7. omplete the flow proof for Theorem - using the following steps. Then write the reasons for each step. a. / and / are supplementary.. / > / c. / 6 m d. / and / are supplementary. Theorem -: If a transversal intersects two parallel lines, then same side interior angles are supplementary. iven: / 6 m Prove: / and / are supplementary. t m,, m iven corr. and are supplementary. linear pair and are supplementary. Sustitution 8. Write a flow proof for the following: a iven: / > / Prove: a 6 iven Vertical are. Sustitution a If corresponding, then lines are parallel. opyright y Pearson Education, Inc., or its affiliates. All Rights Reserved. 0

7 Name lass ate -4 Reteaching Parallel and Perpendicular Lines You can use angle pairs to prove that lines are parallel. The postulates and theorems you learned are the asis for other theorems aout parallel and perpendicular lines. Theorem -8: Transitive Property of Parallel Lines a If two lines are parallel to the same line, then they are parallel to each other. If a 6 and 6 c, then a 6 c. Lines a,, and c can e in c different planes. Theorem -9: If two lines are perpendicular to the same line, then those two lines are parallel to each other. This is only true if all the lines are in the same plane. If a ' d and ' d, then a 6. Theorem -0: Perpendicular Transversal Theorem a d If a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other line. a This is only true if all the lines are in the same plane. If a 6 and c, and a ' d, then ' d, and c ' d. c d. omplete this paragraph proof of Theorem -8. iven: d 6 e, e 6 f Prove: d 6 f Proof: ecause it is given that d 6 e, then / is supplementary to / y the Same-Side Int. Angles Postulate. ecause it is given that e 6 f, then / > / y the orresponding Angles Theorem. So, y sustitution, / is supplementary to /. y the onverse of the Same-Side Int. Angles Postulate, d 6 f. h d e f. Write a paragraph proof of Theorem -9. iven: t ' n, t ' o Prove: n 6 o iven that t ' n and t ' o, ml 5 90 and ml 5 90, y def. of perpendicular lines. Thus l Ol. So, n n o ecause of the onverse of the orr. ' Thm. n o t opyright y Pearson Education, Inc., or its affiliates. All Rights Reserved. 9

8 Name lass ate -4 Reteaching (continued) Parallel and Perpendicular Lines A carpenter is uilding a cainet. A decorative door will e set into an outer frame. 5 a. If the lines on the door are perpendicular to the top of the outer frame, what must e true aout the lines?. The outer frame is made of four separate pieces of molding. Each piece has angled corners as shown. When the pieces are fitted together, will each set of sides e parallel? Explain c. According to Theorem -9, lines that are perpendicular to the same line are parallel to each other. So, since each line is perpendicular to the top of the outer frame, all the lines are parallel. 5 The angles for the top and ottom pieces are 58. The angles for the sides are 558. etermine whether each set of sides will e parallel. raw the pieces as fitted together to determine the measures of the new angles formed. Use this to decide if each set of sides will e parallel. The new angle is the sum of the angles that come together. Since , the pieces form right angles. Two lines that are perpendicular to the same line are parallel. So, each set of sides is parallel.. An artist is uilding a mosaic. The mosaic consists of the repeating pattern shown at the right. What must e true of a and to ensure that the sides of the mosaic are parallel? a 5 50 and a 40 a Error Analysis A student says that according to Theorem -0, if * A ) 6 * ) and * A ) ' * A ), then * ) ' * A ). Explain the student s error. * ) * ) A and are in different planes. E A opyright y Pearson Education, Inc., or its affiliates. All Rights Reserved. 40

9 Name lass ate -5 Reteaching Parallel Lines and Triangles Triangle Angle-Sum Theorem: The measures of the angles in a triangle add up to 80. In the diagram at the right, na is a right triangle. What are m/ and m/? A Step m/ m/a 5 90 Angle Addition Postulate m/ Sustitution Property m/ 5 60 Sutraction Property of Equality Step m/ m/ m/a 5 80 Triangle Angle-Sum Theorem 60 m/ Sustitution Property m/ Addition Property of Equality m/ 5 60 Sutraction Property of Equality 0 60 ind ml Algera ind the value of each variale X Y Z 6 7 7; 6; 4; 8; 6 X Y Z 8 7 X Y Z 90; 90; 58 opyright y Pearson Education, Inc., or its affiliates. All Rights Reserved. 49

10 Name lass ate -5 Reteaching (continued) Parallel Lines and Triangles In the diagram at the right, / is an exterior angle of the triangle. An exterior angle is an angle formed y one side of a polygon and an extension of an adjacent side. or each exterior angle of a triangle, the two interior angles that are not next to it are its remote interior angles. In the diagram, / and / are remote interior angles to /. The Exterior Angle Theorem states that the measure of an exterior angle is equal to the sum of its remote interior angles. So, m/ 5 m/ m/. What are the measures of the unknown angles? m/ A m/a m/a 5 80 Triangle Angle-Sum Theorem A 45 m/ 5 80 Sustitution Property m/ 5 04 Sutraction Property of Equality m/a m/a 5 m/ Exterior Angle Theorem 45 5 m/ Sustitution Property 76 5 m/ Sutraction Property of Equality 45 What are the exterior angle and the remote interior angles for each triangle? 0... P O M L 4 K M N J exterior: l4 exterior: lnop exterior: ljlm interior: l, l interior: lomn, lmno interior: ljkl, lljk ind the measure of the exterior angle U X V T E 46 7 W 5 opyright y Pearson Education, Inc., or its affiliates. All Rights Reserved. 50

11 Name lass ate -6 Reteaching onstructing Parallel and Perpendicular Lines Parallel Postulate Through a point not on a line, there is exactly one line parallel to the given line. iven: Point not on * ) onstruct: * J ) parallel to * ) Step raw * ). Step With the compass tip on, draw an arc that intersects * ) etween and. Lael this intersection point. ontinue the arc to intersect * ) at point. Step Without changing the compass setting, place the compass tip on and draw an arc that intersects * ) aove and. Lael this intersection point. Step 4 Place the compass tip on and open or close the compass so it reaches. raw a short arc at. Step 5 Without changing the compass setting, place the compass tip on and draw an arc that intersects the first arc drawn from. Lael this intersection point J. J Step 6 raw * J ), which is the required line parallel to * ). J opyright y Pearson Education, Inc., or its affiliates. All Rights Reserved. 59

12 Name lass ate -6 Reteaching (continued) onstructing Parallel and Perpendicular Lines onstruct a line parallel to line m and through point Y. Sample answers shown. Z. Y.. m Y Z Y m Z m Perpendicular Postulate Through a point not on a line, there is exactly one line perpendicular to the given line. iven: Point not on * ) onstruct: a line perpendicular to * ) through Step onstruct an arc centered at that intersects * ) at two points. Lael those points and. Step onstruct two arcs of equal length centered at points and. Step onstruct the line through point and the intersection of the arcs from Step. Step Step Step onstruct a line perpendicular to line n and through point X. Sample answers shown. 4. X n X n X n opyright y Pearson Education, Inc., or its affiliates. All Rights Reserved. 60