Ways to Prove Parallel Lines Exist in Geometry
Learn five methods to prove the existence of parallel lines in geometry, including identifying pairs of angles and determining parallel lines based on angle measures. Practice proofs and understand the relationships between angles to establish parallelism.
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Geometry/Trig Unit 3 Review Packet Answer Key Name: __________________________ Date: ___________________________ Section I Name the five ways to prove that parallel lines exist. 1. If corresponding angles are congruent, then lines are parallel. 2. If alternate interior angles are congruent, then lines are parallel. 3. If alternate exterior angles are congruent, then lines are parallel. 4. If same side interior angles are supplementary, then lines are parallel. 5. If same side exterior angles are supplementary, then lines are parallel. Section II Identify the pairs of angles. If the angles have no relationship, write none. 1. 7& 11 None 2. 3& 6 Alternate Interior Angles 3. 8& 16 Corresponding Angles 4. 2& 7 Alternate Exterior Angles 5. 3& 5 Same Side Interior Angles 6. 1& 6 None 7. 1& 6 None 8. 1& 4 Vertical Angles 1 2 9 10 1112 a 3 4 5 6 8 13 14 16 b 7 15 Section III Fill In Vertical angles are congruent. If lines are parallel, then corresponding angles are congruent. If lines are parallel, then alternate interior angles are congruent. If lines are parallel, then alternate exterior angles are congruent. If lines are parallel, then same side interior angles are supplementary. If lines are parallel, then same side exterior angles are supplementary.
Geometry/Trig Unit 3 Review Packet Page 2 Answer Key Name: __________________________ Date: ___________________________ Section IV Determine which lines, if any, are parallel based on the given information. 1.) m 1 = m 9 c // d 1 2 9 10 1112 a 3 4 2.) m 1 = m 4 None 5 6 8 13 14 16 b 3.) m 12 + m 14 = 180 a // b 7 15 4.) m 1 = m 13 None c d 5.) m 7 = m 14 c // d 6.) m 13 = m 11 None 7.) m 15 + m 16 = 180 None 8.) m 4 = m 5 a //b Section IV Determine which lines, if any, are parallel based on the given information. 1. m 1 = m 4 a // b 2. m 6 = m 8 t // s 3. 1 and 11 are supplementary None 4. a t and b t a // b a 5. m 14 = m 5 None b k m 6. 6 and 7 are supplementary t // s 15 t 13 12 11 9 8 7. m 14 = m 15 k // m 7 10 8. 7 and 8 are supplementary None 5 2 1 3 4 6 s 9. m 5 = m 10 k // m 14 10. m 1 = m 13 None
Geometry/Trig Unit 3 Review Packet Page 3 Answer Key Name: __________________________ Date: ___________________________ J Section V - Proofs 1. Given: GK bisects JGI; m 3 = m 2 Prove: GK // HI 1 G K 2 Statements Reasons 1. Given 1. GK bisects JGI 3 I 2. Definition of an Angles Bisector 2. m 1 = m 2 H 3. m 3 = m 2 3. Given 4. m 1 = m 3 4. Substitution 5. GK // HI 5. If corresponding angles are congruent, then the lines are parallel. A C 2. Given: AJ // CK; m 1 = m 5 Prove: BD // FE Reasons Statements 1 2 3 1. AJ // CK 1. Given B D 4 2. m 1 = m 3 2. If lines are parallel, then corresponding angles are congruent. 5 F E J K 3. m 1 = m 5 3. Given 4. m 3 = m 5 4. Substitution 5. BD // FE 5. If corresponding angles are congruent, then the lines are parallel.
Geometry/Trig Unit 3 Review Packet Page 4 Answer Key Name: __________________________ Date: ___________________________ 3. Given: a // b; 3 4 Prove: 10 1 1 2 a 3 4 5 Statements Reasons 6 1. a // b 1. Given 8 7 b 2. 4 7 2. If lines are parallel then alternate interior angles are congruent. 9 10 d c 3. 3 4 4. 3 7 5. 1 3; 7 10 6. 10 1 3. Given 4. Substitution 5. Vertical Angles Theorem 6. Substitution 4. Given: 1 and 7 are supplementary. Prove: m 8 = m 4 1 3 b 4 5 6 7 a 8 2 Statements Reasons 1. 1 and 7 are supplementary 2. m 1 + m 7 = 180 3. m 6 + m 7 = 180 1. Given 2. Definition of Supplementary Angles 3. Angle Addition Postulate 4. m 1 + m 7 = m 6 + m 7 4. Substitution 5. m 1 = m 6 5. Subtraction Property 6. a // b 6. If corresponding angles are congruent, then the lines are parallel. 7. m 8 = m 4 7. If lines are parallel, then corresponding angles are congruent.
Geometry/Trig Unit 3 Review Packet Page 5 Answer Key Name: __________________________ Date: ___________________________ 5. Given: ST // QR; 1 3 Prove: 2 3 P Reasons Statements 1. ST // QR 1. Given 1 3 S T 2. 1 2 2. If lines are parallel, then corresponding angles are congruent. 2 Q R 3. 1 3 3. Given 4. 2 3 4. Substitution 6. Given: BE bisects DBA; 1 3 Prove: CD // BE Reasons Statements 1. BE bisects DBA 1. Given 2. 2 3 2. Definition of an Angle Bisector 3. 1 3 3. Given 4. 2 1 4. Substitution 5. CD // BE 5. If alternate interior angles are congruent, then the lines are parallel. C B 2 3 1 A D E
Geometry/Trig Unit 3 Review Packet page 6 Answer Key Name: __________________________ Date: ___________________________ 7. Given: AB // CD; BC // DE Reasons Statements Prove: 2 6 1. AB // CD 1. Given 2. 2 4 2. If lines are parallel, then alternate interior angles are congruent. 3. BC // DE 3. Given 4. 4 6 4. If lines are parallel, then alternate interior angles are congruent. 5. 2 6 5. Substitution B D 6 2 4 1 3 5 7 A C E 8. Given: AB // CD; 2 6 Prove: BC // DE Reasons Statements 1. AB // CD 1. Given 2. 2 4 2. If lines are parallel, then alternate interior angles are congruent. 3. 2 6 3. Given 4. 4 6 4. Substitution 5. BC // DE 5. If alternate interior angles are congruent, then the lines are parallel. B D 6 2 4 1 3 5 7 A C E
Geometry/Trig Unit 3 Review Packet page 7 Answer Key Name: __________________________ Date: ___________________________ Section VI Solve each Algebra Connection Problem. 1. w 2. 4x - 5 23y z + 57 x 65 125 37 2y w = 37 x = 143 y = 71.5 z = 86 Equations: 37 = w x + 37 = 180 2y + 37 = 180 z + 57 = 143 Equations: 65 + 23y = 180 65 = 4x 5 x = 17.5 y = 5 Equations: 30 + 75 = 5x 30 + 75 + y = 180 Equation: 6x + x + 12 = 8x + 1 3. 4. 30 x + 12 5x y 75 6x 8x + 1 x = 21 y = 75 x = 11 Section VII Determine whether the given side lengths can create a triangle. 1) 7, 8, 9 YES 2) 7, 8, 15 NO 3) 7, 8, 14 YES 4) 3, 4, 5 YES
Geometry/Trig Unit 3 Review Packet page 8 Name: __________________________ Date: ___________________________ Section VIII - Classify each triangle by its sides and by its angles. 2. D 3. G 1. A 47 53 E H F I B C Scalene Right Scalene Acute Scalene Obtuse K 4. 5. 6. O Q 32 P 118 36 J M 60 L R N Equilateral Equiangular Scalene Obtuse Scalene Acute 7. In ABC which side is the longest? ___BC_____ the shortest? ______AC___ 8. In DEF which side is the longest? _DE___ the shortest? ___EF_______ 9. In GHI which side is the longest? ___HI_____ the shortest? _____GI____ 10. In JKL which side is the longest? ____JK____ the shortest? ____KL_____ 11. In MNO which side is the longest? ____all the same____ 12. In PQR which side is the longest? _QR____ the shortest___PR_____ In each triangle, name the smallest angle and the largest angle. A D I 6.4 4.1 128 17.3 F C G 5.7 B E 136 H 11 Smallest <B Smallest <E Smallest <I Largest <C Largest <D Largest <G
