Basic Quadrilateral Proofs


 Cleopatra Norman
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1 Basic Quadrilateral Proofs For each of the following, draw a diagram with labels, create the givens and proof statement to go with your diagram, then write a twocolumn proof. Make sure your work is neat and organized. Quadrilateral Proof: 1. Prove that the sum of the interior angles of a quadrilateral is 360 o. Given: Quadrilateral ABCD Prove: m A + m B + m C + m D = 360 o 1. Quadrilateral ABCD 1. Given 2. m BAC + m B + m BCA = 180 o m CAD + m D + m ACD = 180 o 2. The sum of the interior angles of a triangles add up to 180 o 3. m BAC + m B + m BCA + 3. Addition and Substitution postulates m CAD + m D + m ACD = 360 o 4. m A = m BAC + m CAD 4. Angle addition postulate m C = m BCA + m ACD 5. m A + m B + m C + m D = 360 o 5. Substitution postulate (steps 5,6) Parallelogram Proofs: 2. Prove the opposite sides of a parallelogram are congruent. Prove: AB CD and BC AD 2. AB CD and BC AD 2. Definition of parallelogram 3. BAC DCA and 3. Alternate Interior angles are DAC BCA 4. AC AC 4. Reflexive property of 5. ΔABC ΔCDA 5. ASA 6. AB CD and BC AD 6. CPCTC
2 3. Prove that any pair of consecutive angles of a parallelogram are supplementary. Prove: A and B are supplementary 2. BC AD 2. Definition of parallelogram 3. A and B are supplementary 3. Same side interior angles are supplementary. 4. Prove that opposite angles of a parallelogram are congruent. Prove: A C and B D 2. AB CD and BC AD 2. Definition of parallelogram 3. BAC DCA and DAC 3. Alternate Interior angles are BCA 4. AC AC 4. Reflexive property of 5. ΔABC ΔCDA 5. ASA 6. B D 6. CPCTC 7. A and B are supplementary 7. Property of parallelogram proved in problem #3 C and D are supplementary 8. B D 8. Supplements of angles are (steps 6,7) 5. Prove that the diagonals of a parallelogram bisect each other. Prove: AE CE and BE DE 2. AB CD 2. Definition of parallelogram 3. ABE CDE 3. Alternate Interior Angles are (step 2) BAE DCE 4. AB CD 4. Property of parallelogram proved in #2 5. ΔAEB ΔCED 5. ASA 6. AE CE and BE DE 6. CPCTC
3 Rhombus Proofs: 6. Prove that a rhombus is a parallelogram. Given: Rhombus ABCD Prove: AB CD and BC AD 1. Rhombus ABCD 1. Given 2. AB CD BC AD 2. Definition of rhombus 3. AC AC 3. Reflexive property of 4. ΔABC ΔCDA 4. SSS 5. BAC DCA 5. CPCTC 6. AB CD 6. Converse of alternateinterior angles 7. CAD ACB 7. CPCTC 8. BC AD 8. Converse of alternateinterior angles 7. Prove that the diagonals of a rhombus are perpendicular. Given: Rhombus ABCD Prove: AC BD 1. Rhombus ABCD 1. Given 2. AB BC 2. Definition of rhombus 3. ABCD is a parallelogram 3. Property of a rhombus proved in #6 4. AE CE 4. Property of a parallelogram proved in #5 5. ΔABE ΔCBE 5. SSS 6. AEB CEB 6. CPCTC 7. m AEB = m CEB 7. Definition of segment 8. AEB and CEB are supplementary 8. Two angles that form a linear pair are supplementary 9. m AEB + m CEB = 180 o 9. Definition of supplementary angles 10. m AEB + m AEB = 180 o 10. Substitution postulate (steps 7,9) 11. m AEB = 90 o 11. Division postulate 12. AEB is a right angle 12. Definition of right angle 13. AC BD 13. Definition of perpendicular segments
4 8. Prove that each diagonal of a rhombus bisects a pair of opposite angles. Given: Rhombus ABCD Prove: BAE DAE, BCE DCE, ABE CBE, ADE CDE 1. Rhombus ABCD 1. Given 2. AB CD BC AD 2. Definition of rhombus 3. ABCD is a parallelogram 3. Property of rhombus proved in #6 4. AE CE and BE DE 4. Property of parallelogram proved in #5 5. ΔAEB ΔCEB ΔCED ΔAED 5. SSS 6. BAE DAE, BCE DCE, 6. CPCTC ABE CBE, ADE CDE Rectangles Proofs: 9. Prove that a rectangle is a parallelogram. Given: Rectangle ABCD Prove: AB CD and BC AD 1. Rectangle ABCD 1. Given 2. A B C D 2. Definition of rectangle 3. m A = m B = m C = m D 3. Definition of angles 4. m A + m B + m C + m D = 360 o 4. Property of Quadrilateral proved in # m A = 360 o 5. Substitution postulate (steps 3,7) 6. m A = 90 o 6. Division postulate 7. m B = m C = m D = 90 o 7. Substitution postulate (steps 3,9) 8. m A + m B = 180 o 8. Substitution postulate m A + m D = 180 o 9. A and B are supplementary 9. Definition of supplementary angles A and D are supplementary 10. AB CD and BC AD 10. Converse of sameside interior
5 10. Prove that the diagonals of a rectangle are congruent. Given: Rectangle ABCD Prove: AC BD 1. Rectangle ABCD 1. Given 2. A D 2. Definition of rectangle 3. AD AD 3. Reflexive Property 4. ABCD is parallelogram 4. Property of rectangle proved in #9 5. AB CD 5. Property of parallelogram proved in #2 6. ΔBAD ΔCDA 6. SAS 7. AC BD 7. CPCTC Trapezoid Proofs: 11. Prove each pair of base angles of an isosceles trapezoid is congruent. Given: Isosceles Trapezoid ABCD with BC AD Prove: A D and ABC C 1. Isosceles Trapezoid ABCD with BC AD 1. Given 2. AB CD 2. Definition of Isosceles Trapezoid 3. Construct BE CD 3. Construction 4. Parallelogram EBCD 4. Definition of parallelogram (steps 1,3) 5. BE CD 5. Property of parallelogram proved in #2 6. AB BE 6. Transitive property of (steps 2,5) 7. A AEB 7. Isosceles triangle theorem 8. AEB D 8. Corresponding angles are 9. A D 9. Transitive property 10. A and ABC are supplementary 10. Same side interior angles are supplementary C and D are supplementary 11. ABC C 11. Supplements of angles are (steps 9,10)
6 11. Prove that the diagonals of an isosceles trapezoid are congruent. Given: Isosceles Trapezoid ABCD Prove: AC BD 1. Isosceles Trapezoid ABCD with BC AD 1. Given 2. AB CD 2. Definition of Isosceles Trapezoid 3. BAD CDA 3. Property of Isosceles Trapezoid proved in #11 4. AD AD 4. Reflexive Property 5. ΔBAD ΔCDA 5. SAS 6. AC BD 6. CPCTC Kite Proofs: 12. Prove that the diagonals of a kite are perpendicular. Given: Kite ABCD Prove: AC BD 1. Kite ABCD 1. Given 2. AB BC, AD CD 2. Definition of Kite 3. BAC BCA 3. Isosceles Triangle theorem 4. BD BD 4. Reflexive property of 5. ΔDAB ΔDCB 5. SSS (steps 2,4) 6. ABD CBD 6. CPCTC 7. ΔABE ΔCBE 7. ASA (steps 2,3,6) 8. AEB CEB 8. CPCTC 9. AEB and CEB are supplementary 9. Two angles that form a linear pair are supplementary 10. m AEB + m CEB = 180 o 10. Definition of supplementary 11. m AEB = m CEB 11. Definition of angles (step 8) 12. m AEB + m AEB = 180 o 12. Substitution property (steps 10,11) 13. m AEB = 90 o 13. Division property 14. AC BD 14. Definition of perpendicular segments
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