Lesson 6.1 Skills Practice Name Date Time to Get Right Right Triangle Congruence Theorems Vocabulary Choose the diagram that models each right triangle congruence theorem. 1. Hypotenuse-Leg (HL) Congruence Theorem a. X Y R Q P Z 2. Leg-Leg (LL) Congruence Theorem b. U V W X 3. Hypotenuse-Angle (HA) Congruence Theorem c. E F © Carnegie Learning G I 4. Leg-Angle (LA) Congruence Theorem H d. 6 W T U X V Y Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 587 587 17/06/13 11:37 AM Lesson 6.1 Skills Practice page 2 Problem Set Mark the appropriate sides to make each congruence statement true by the Hypotenuse‑Leg Congruence Theorem. 1. DPR QFM R 2.ACI GCE A M I C F D E Q P G 3. QTR SRT T 4.ADG HKN Q D A S N K R H G Mark the appropriate sides to make each congruence statement true by the Leg‑Leg Congruence Theorem. 5. BZN TGC 6.MNO QPO G T N P B O 6 Z C 7. PZT PZX M Q 8.EGI ONQ P E G N I Q O © Carnegie Learning N T Z X 588 Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 588 17/06/13 11:37 AM Lesson 6.1 Skills Practice page 3 Name Date Mark the appropriate sides and angles to make each congruence statement true by the Hypotenuse‑Angle Congruence Theorem. 9. SVM JFW 10.MSN QRT W F V P J S N T M S 11. IEG IEK M Q R 12.DCB ZYX E G Y D B I X K C Z Mark the appropriate sides and angles to make each congruence statement true by the Leg‑Angle Congruence Theorem. 13. XTD HPR 14.SEC PEC P X T S R © Carnegie Learning T C D D E 6 H P R Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 589 589 17/06/13 11:37 AM Lesson 6.1 Skills Practice 15. PBJ OTN page 4 16.AXT YBU P A X B Y B J T O T U N For each figure, determine if there is enough information to prove that the two triangles are congruent. If so, name the congruence theorem used. ___ ___ ___ 17. Given: GF bisects /RGS, and 18. Given: DV TU /R and /S are right angles. Is nDVT > nDVU? Is nFRG > nFSG? G D S R T V U F es. There is enough information Y to conclude that nFRG > nFSG by HA. ____ ____ ____ ____ ___ ___ 19. Given: NM > EM , DM > OM , and 20. Given: RP > QS , and /R and /Q /NMD and /EMO are right angles. are right angles. Is nNMD > nEMO? Is nSRP > nPQS? N O 6 M S P Q D E 590 R © Carnegie Learning Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 590 17/06/13 11:37 AM Lesson 6.1 Skills Practice page 5 Name Date ____ ____ ___ ____ 21. Given: GO > MI , and /E and /K are 22. Given: HM > VM , and /H and /V are right angles. right angles. Is nGEO > nMKI? Is nGHM > nUVM? E G I G H M O M K V U Use the given information to answer each question. 23. T wo friends are meeting at the library. Maria leaves her house and walks north on Elm Street and then east on Main Street to reach the library. Paula leaves her house and walks south on Park Avenue and then west on Main Street to reach the library. Maria walks the same distance on Elm Street as Paula walks on Main Street, and she walks the same distance on Main Street as Paula walks on Park Avenue. Is there enough information to determine whether Maria’s walking distance is the same as Paula’s walking distance? Paula’s house W Park Avenue N E S Elm Street © Carnegie Learning Main Street Library Maria’s house Yes. Maria’s walking distance to the library is equal to Paula’s walking distance. The triangles formed are right triangles. The corresponding legs of the triangles are congruent. So, by the Leg-Leg Congruence Theorem, the triangles are congruent. If the triangles are congruent, the hypotenuses are congruent. Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 591 6 591 17/06/13 11:37 AM Lesson 6.1 Skills Practice page 6 24. A n auto dealership displays one of their cars by driving it up a ramp onto a display platform. Later they will drive the car off the platform using a ramp on the opposite side. Both ramps form a right triangle with the ground and the platform. Is there enough information to determine whether the two ramps have the same length? Explain. Display platform Ground 25. A radio station erected a new transmission antenna to provide its listeners with better reception. The antenna was built perpendicular to the ground, and to keep the antenna from swaying in the wind two guy wires were attached from it to the ground on opposite sides of the antenna. Is there enough information to determine if the guy wires have the same length? Explain. guy wire Tower guy wire © Carnegie Learning Ground 6 592 Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 592 17/06/13 11:37 AM Lesson 6.1 Skills Practice page 7 Name Date 26. T wo ladders resting on level ground are leaning against the side of a house. The bottom of each ladder is exactly 2.5 feet directly out from the base of the house. The point at which each ladder rests against the house is 10 feet directly above the base of the house. Is there enough information to determine whether the two ladders have the same length? Explain. Create a two-column proof to prove each statement. ____ ___ ____ ___ ___ ___ 27. Given: WZ bisects VY , WV VY , and YZ VY W Prove: WVX ZYX X Y V © Carnegie Learning Z ____ ___ Statements ___ ___ 1. WV VY and YZ VY Reasons 1. Given 2. WVX and ZYX are right angles. 2. Definition of perpendicular angles 3. WVX and ZYX are right triangles. ____ ___ 4. WZ bisects VY ___ ___ . 5. VX YX 3. Definition of right triangles 6. WXV ZXY 6. Vertical Angle Theorem 7. WVX ZYX 7. LA Congruence Theorem 4. Given 5. Definition of segment bisector Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 593 6 593 17/06/13 11:37 AM Lesson 6.1 Skills Practice page 8 ___ 28. G iven: Point D is the midpoint of EC , A ADB is an isosceles triangle with base ___ AB , and E and C are right angles. Prove: AED BCD ___ ___ ___ E ___ ___ ___ , TP UP , and UR PR 29. Given: SU UP B D T C P Prove: SUR TPR R 6 594 U S © Carnegie Learning Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 594 17/06/13 11:37 AM Lesson 6.1 Skills Practice page 9 Name Date 30. Given: Rectangle MNWX and NMW XWM M Prove: MNW WXM X W © Carnegie Learning N 6 Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 595 595 17/06/13 11:37 AM © Carnegie Learning 6 596 Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 596 17/06/13 11:37 AM Lesson 6.2 Skills Practice Name Date CPCTC Corresponding Parts of Congruent Triangles are Congruent Vocabulary Provide an example to illustrate each term. 1. Corresponding parts of congruent triangles are congruent (CPCTC) 2. Isosceles Triangle Base Angle Theorem © Carnegie Learning 3. Isosceles Triangle Base Angle Converse Theorem 6 Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 597 597 17/06/13 11:37 AM Lesson 6.2 Skills Practice page 2 Problem Set Create a two-column proof to prove each statement. ___ ___ 1. Given: RS is the bisector of PQ . Q Prove: SPT SQT T R P Statements ___ ___ 1. RS is the bisector of PQ . ___ ___ 2. RS PQ Reasons 1. Given 3. PTS and QTS are right angles. 3. Definition of perpendicular lines 4. PTS ___ and QTS ___ are right triangles. 4. Definition of right triangles 2. Definition of perpendicular bisector 5. RS bisects PQ ___ ___ ___ ___ 7. TS TS 5. Definition of perpendicular bisector 8. PTS QTS 8. Leg-Leg Congruence Theorem 9. SPT SQT 9. CPCTC 6. PT QT ___ ____ ___ 6. Definition of bisect 7. Reflexive Property of ____ ___ ____ , TM WT , and TZ WX 2. Given: TZ WX M ____ ___ Prove: MZ TX T Z X 6 598 W © Carnegie Learning S Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 598 17/06/13 11:37 AM Lesson 6.2 Skills Practice page 3 Name Date ___ ___ 3. G ___ iven:___ AG ___ and EK ____ intersect at C, AC EC , CK CG A E C Prove: K G K G 4. Given: JHK LHK, JKH LKH ___ H ___ Prove: JK LK L J K © Carnegie Learning 6 Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 599 599 17/06/13 11:37 AM Lesson 6.2 Skills Practice page 4 5. Given: UGT SGB B T Prove: TUS BSU G S U ___ ___ 6. Given: TPN TNP, TP QP P Prove: ___ ___ TN QP T 6 600 N © Carnegie Learning Q Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 600 17/06/13 11:37 AM Lesson 6.2 Skills Practice page 5 Name Date ___ ___ ___ ___ 7. Given: AC DB , AC bisects DB A ___ ___ Prove: AD AB E D C ___ ___ ___ ___ , FK JK 8. Given: KGH KHG, FG JH © Carnegie Learning G H Prove: F J F B K J 6 Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 601 601 17/06/13 11:37 AM Lesson 6.2 Skills Practice ___ page 6 ___ ___ , AC bisects TAQ 9. Given: AT AQ T ___ ___ Prove: AC bisects TQ C Q A ___ __ __ , LNJ IGJ, J is the midpoint of LI 10. Given: EL EI E ___ ___ Prove: NJ GJ N L I 6 602 J © Carnegie Learning G Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 602 17/06/13 11:37 AM Lesson 6.2 Skills Practice page 7 Name Date 11. Given: E EUV, F FVU E ___ ___ Prove: UF VE U V F ___ ___ ___ ___ 12. Given: CT CP , AT AP C Prove: mCTA 5 mCPA A T © Carnegie Learning P 6 Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 603 603 17/06/13 11:37 AM Lesson 6.2 Skills Practice page 8 Use the given information to answer each question. 13. S amantha is hiking through the forest and she comes upon a canyon. She wants to know how wide the canyon is. She measures the distance between points A and B to be 35 feet. Then, she measures the distance between points B and C to be 35 feet. Finally, she measures the distance between points C and D to be 80 feet. How wide is the canyon? Explain. A D B E C The canyon is 80 feet wide. he triangles are congruent by the Leg-Angle T ____ ___Congruence Theorem. Corresponding parts of congruent triangles are congruent, so CD 5 AE . 14. Explain why mNMO 5 20°. M 60° Q 6 80° P O N 15. Calculate MR given that the perimeter of HMR is 60 centimeters. M 60° 20 cm © Carnegie Learning 60° H 604 R Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 604 17/06/13 11:37 AM Lesson 6.2 Skills Practice Name page 9 Date 16. Greta has a summer home on Lake Winnie. Using the diagram, how wide is Lake Winnie? Lake Winnie 52 m 20 m 20 m Greta’s summer home 52 m 48 m 17. J ill is building a livestock pen in the shape of a triangle. She is using one side of a barn for one of the sides of her pen and has already placed posts in the ground at points A, B, and C, as shown in the diagram. If she places fence posts every 10 feet, how many more posts does she need? Note: There will be no other posts placed along the barn wall. A Barn wall Livestock pen B 50' © Carnegie Learning C 6 18. Given rectangle ACDE, calculate the measure of CDB. A B C 30° E D Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 605 605 17/06/13 11:37 AM © Carnegie Learning 6 606 Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 606 17/06/13 11:37 AM Lesson 6.3 Skills Practice Name Date Congruence Theorems in Action Isosceles Triangle Theorems Vocabulary Choose the term from the box that best completes each sentence. Isosceles Triangle Altitude to Congruent Sides Theorem Isosceles Triangle Base Theorem Isosceles Triangle Vertex Angle Theorem Isosceles Triangle Angle Bisector to Congruent Sides Theorem Isosceles Triangle Perpendicular Bisector Theorem 1. A (n) two congruent legs in an isosceles triangle. vertex angle is the angle formed by the 2. In an isosceles triangle, the altitudes to the congruent sides are congruent, as stated in the . 3. In an isosceles triangle, the angle bisectors to the congruent sides are congruent, as stated in the . © Carnegie Learning 4. T he states that the altitude from the vertex angle of an isosceles triangle is the perpendicular bisector of the base. 5. T he the base of an isosceles triangle bisects the base. states that the altitude to 6. The altitude to the base of an isosceles triangle bisects the vertex angle, as stated in the . Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 607 6 607 17/06/13 11:37 AM Lesson 6.3 Skills Practice page 2 Problem Set Write the theorem that justifies the truth of each statement. ___ ____ ___ 1. In isosceles MRG, RD GC . 2.___ In isosceles TGC with altitude TP , ____ ___ ___ TP GC , and GP CP . G R C T G D P M C Isosceles Triangle Angle Bisector to Congruent Sides Theorem ___ ___ 3. In isosceles BRU with altitude BD , 4. In isosceles JFI with altitude JH , ___ ___ UD RD . /HJF /HJI. B I H R F D U J ___ ___ O K N M 6 B A H M © Carnegie Learning 5. In isosceles nMNO, OA > NB. 6. In isosceles nHJK, KN ___ bisects /HKJ, ___ ___ JM bisects /HJK, and MJ > NK . N J 608 Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 608 17/06/13 11:37 AM Lesson 6.3 Skills Practice page 3 Name Date Determine the value of x in each isosceles triangle. 7. B 8 in. S 25 m 32 ° x° D 8. K 24 m 8 in. A 25 m P C x J W x 5 32° 9. 10. M 20 ft P x S 4m V 20 ft D 26 ft x U N 16 m 16 m 20° 20° T © Carnegie Learning 11. W 12. P Q 10 cm V 29 yd x 12 cm 12 cm 37° x R 29 yd T U Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 609 6 609 17/06/13 11:37 AM Lesson 6.3 Skills Practice page 4 Complete each two-column proof. ___ ___ AB CB , 13. G ___ iven:___ Isosceles ABC with ___ ___ ___ ___ BD AC , DE AB , and DF CB ___ B ___ E Prove: ED FD A D ___ ____ Statements 1. AB CB ___ ___ ____ ___ ___ ____ 2. BD AC , DE AB , DF CB 2. Given 3. AED and CFD are right angles. 3. Definition of perpendicular lines 4. AED and CFD are right triangles. 4. Definition of right triangle 5. A C 5. Base Angle Theorem C Reasons 1. Given 6. Isosceles Triangle Base Theorem 7. HA Congruence Theorem 8. CPCTC © Carnegie Learning ___ ____ 6. AD CD 7. AED CFD ___ ___ 8. ED FD 6 610 F Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 610 17/06/13 11:37 AM Lesson 6.3 Skills Practice page 5 Name Date ____ ____ 14.Given: Isosceles MNB with MN MB , ____ ___ NO bisects ANB, BA bisects OBN M Prove: BAN NOB B A N © Carnegie Learning O 6 Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 611 611 17/06/13 11:37 AM Lesson 6.3 Skills Practice __ __ ___ page 6 __ ___ __ 15. Given: Isosceles IAE with IA IE , AG IE , EK IA A E Prove: IGA IKE M K G I ___ ____ ____, 16. Given: Isosceles GQR with GR ____ GQ Isosceles QGH____ with GQ ___ QH ___ , ___ ___ ___ GJ QR , QP GH , and GJ QP ___ G P J Q H ___ Prove: RJ HP R © Carnegie Learning 6 612 Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 612 17/06/13 11:37 AM Lesson 6.3 Skills Practice page 7 Name Date Use the given information to answer each question. 17. T he front ___ of an A-frame house is in the shape of an isosceles triangle, as shown in the diagram. In the ___ ____ ___ diagram, HK GJ , GH JH , and mHGJ 5 68.5°. Use this information to determine the measure of GHJ. Explain. H G K J The measure of GHJ is 43°. By the Triangle Sum Theorem, mGHK 5 180° 2 (90° 1 68.5°) 5 21.5°. y the Isosceles Triangle Vertex Angle Theorem, mGHK 5 mJHK. B Therefore, mGHJ 5 21.5° 1 21.5° 5 43°. 18. W hen building a house, rafters are used to support the___ roof. The rafter shown in the diagram ___ ___ ____ has the shape of an isosceles triangle. In the diagram, NP RQ ____, NR NQ , NP 5 12 feet, and RP 5 16 feet. Use this information to determine the length of NQ . Explain. © Carnegie Learning N R P Q 6 Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 613 613 17/06/13 11:37 AM Lesson 6.3 Skills Practice page 8 19. S tained glass windows are constructed using different pieces of colored glass held together by lead. The stained glass window in the diagram is rectangular with six different colored glass pieces___ represented by TBS, PBS, PBQ, QBR, NBR, and NBT. Triangle TBP with altitude SB and ___ QBN with altitude RB , are congruent isosceles triangles. If the measure of NBR is 20°, what is the measure of STB? Explain. T S P B N R Q 20. W hile growing up, Nikki often camped out in her back yard in a pup tent. A pup tent has two rectangular sides made of canvas, and a front and back in the shape of two isosceles triangles also ____ made of canvas. The zipper in front, represented by ___ MG in the diagram, is the height of ___ the pup tent and the altitude of isosceles EMH. If the length of EG is 2.5 feet, what is the length of HG ? Explain. K M 6 614 E G T H © Carnegie Learning Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 614 17/06/13 11:37 AM Lesson 6.3 Skills Practice page 9 Name Date ___ ___ ____ ___ 21. A beaded purse is in the shape of an isosceles triangle. In the diagram, TN TV , VM TN , ___ ___ ___ and NU TV . How long is the line of beads represented by NU , if TV is 13 inches and TM is 5 inches? Explain. T M R V © Carnegie Learning N U 6 Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 615 615 17/06/13 11:37 AM Lesson 6.3 Skills Practice page 10 22. A kaleidoscope is a cylinder with mirrors inside and an assortment of loose colored beads. When a person looks through the kaleidoscope, different colored shapes and patterns are created as the kaleidoscope is rotated. Suppose that the diagram represents the shapes that a person sees when they look into the kaleidoscope. Triangle AEI is an isosceles triangle with ___ ___ ___ __ ___ AE . EK bisects AEI and IC bisects AIE. What is the length of IC, if one half the length of ___ AI EK is 14 centimeters? Explain. A K C M E © Carnegie Learning I 6 616 Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 616 17/06/13 11:37 AM Lesson 6.4 Skills Practice Name Date Making Some Assumptions Inverse, Contrapositive, Direct Proof, and Indirect Proof Vocabulary Define each term in your own words. 1. inverse 2. contrapositive 3. direct proof 4. indirect proof (or proof by contradiction) © Carnegie Learning 5. Hinge Theorem 6 6. Hinge Converse Theorem Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 617 617 21/06/13 11:39 AM Lesson 6.4 Skills Practice page 2 Problem Set Write the converse of each conditional statement. Then, determine whether the converse is true. 1. If two lines do not intersect and are not parallel, then they are skew lines. The converse of the conditional would be: If two lines are skew lines, then they do not intersect and are not parallel. The converse is true. 2. If two lines are coplanar and do not intersect, then they are parallel lines. 3. If a triangle has one angle whose measure is greater than 90º, then the triangle is obtuse. 6 5. If the lengths of the sides of a triangle measure 5 mm, 12 mm, and 13 mm, then it is a right triangle. 618 © Carnegie Learning 4. If a triangle has two sides with equal lengths, then it is an isosceles triangle. Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 618 17/06/13 11:37 AM Lesson 6.4 Skills Practice page 3 Name Date 6. If the lengths of the sides of a triangle are 3 cm, 4 cm, and 5 cm, then the triangle is a right triangle. 7. If the corresponding sides of two triangles are congruent, then the triangles are congruent. 8. If the corresponding angles of two triangles are congruent, then the triangles are similar. Write the inverse of each conditional statement. Then, determine whether the inverse is true. 9. If a triangle is an equilateral triangle, then it is an isosceles triangle. The inverse of the conditional would be: If a triangle is not an equilateral triangle, then it is not an isosceles triangle. © Carnegie Learning The inverse is not true. 10. If a triangle is a right triangle, then the sum of the measures of its acute angles is 90º. Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 619 6 619 17/06/13 11:37 AM Lesson 6.4 Skills Practice page 4 11. If the sum of the internal angles of a polygon is 180º, then the polygon is a triangle. 12. If a polygon is a triangle, then the sum of its exterior angles is 360º. 13. If two angles are the acute angles of a right triangle, then they are complementary. 14. If two angles are complementary, then the sum of their measures is 90º. 6 16. If a polygon is a trapezoid, then it is a quadrilateral. 620 © Carnegie Learning 15. If a polygon is a square, then it is a rhombus. Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 620 17/06/13 11:37 AM Lesson 6.4 Skills Practice Name page 5 Date Write the contrapositive of each conditional statement. Then, determine whether the contrapositive is true. 17. If one of the acute angles of a right triangle measures 45º, then it is an isosceles right triangle. The contrapositive of the conditional would be: If a triangle is not an isosceles right triangle, then it is not a right triangle with an acute angle that measures 45º. The contrapositive is true. 18. If one of the acute angles of a right triangle measures 30º, then it is a 30º260º290º triangle. 19. If a quadrilateral is a rectangle, then it is a parallelogram. © Carnegie Learning 20. If a quadrilateral is an isosceles trapezoid, then it has two pairs of congruent base angles. 6 Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 621 621 17/06/13 11:37 AM Lesson 6.4 Skills Practice page 6 21. If the sum of the measures of two angles is 180º, then the angles are supplementary. 22. If two angles are supplementary, then the sum of their measures is 180º. 23. If the radius of a circle is 8 meters, then the diameter of the circle is 16 meters. © Carnegie Learning 24. If the diameter of a circle is 12 inches, then the radius of the circle is 6 inches. 6 622 Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 622 17/06/13 11:37 AM Lesson 6.4 Skills Practice page 7 Name Date Create an indirect proof to prove each statement. ____ ____ ____ 25. Given: WY bisects XYZ and XW ZW Y ___ ___ Prove: XY ZY X ___ ___ Statements 1. XY ZY ____ 2. WY bisects XYZ 3. XYW ZYW ____ ____ 4. YW YW 5. XYW ZYW ____ ____ 6. XW ZW ____ ____ 7. XW ZW ___ ___ 8. XY ZY is false. ___ ___ is true. 9. XY ZY Z Reasons 1. Assumption 2. Given 3. Definition of angle bisector 4. Reflexive Property of 5. SAS Congruence Theorem 6. CPCTC 7. Given 8. Step 7 contradicts Step 6. The assumption is false. 9. Proof by contradiction 26. Given: mEBX mEBZ ___ Prove: EB is W X not an altitude of EZX. B E Z © Carnegie Learning 6 Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 623 623 17/06/13 11:38 AM Lesson 6.4 Skills Practice page 8 ___ 27. Given: OMP MOP and NP does not bisect ONM. M ____ ____ Prove: NM NO P N O ___ ___ ___ ___ and EU DU 28. Given: ET DT T ___ ___ Prove: EX DX E X D © Carnegie Learning U 6 624 Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 624 17/06/13 11:38 AM Lesson 6.4 Skills Practice page 9 Name Date For each pair of triangles, use the Hinge Theorem or its converse to write a conclusion using an inequality, 29. P 30. G D R A 108° N S 68° B X 120° F Q Q SP . GQ 31. P 3 32. X R I F 5 T A E Z C 4.5 3.5 U © Carnegie Learning K 6 Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 625 625 17/06/13 11:38 AM © Carnegie Learning 6 626 Chapter 6 Skills Practice 451448_Skills_Ch06_587-626.indd 626 17/06/13 11:38 AM

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