A Given: ABCD is a rhombus. B Prove: ABCD is a parallelogram.
D C
Statement Justification
1. AB ≅ CD & DA ≅ BC . 1. Property of a rhombus. 2. AC ≅ AC . 2. Reflexive axiom. 3. ∆ ACD ≅ ∆ CAB . 3. SSS. 4. ∠ CDA + ( ∠ DAC + ACD ) = 180 o . 4. Interior angle sum for a triangle. 5. ∠ ACD ≅ ∠ CAB . 5. CPCTC 6. ∠ CDA + ( ∠ DAC + ∠ CAB ) = 180 o. 6. Substitution. 7. ∠ CDA & ∠ DAB are supplementary. 7. ( ∠ DAC + ∠ CAB ) = ∠ DAB . 8. CD || AB . 8. Co-interior angles. 9. Repeat 4 – 8 to show BC || DA . A Given: ABCD is a rhombus. B Prove: ABCD is a parallelogram.
D C
Statement Justification
1. AB ≅ CD & DA ≅ BC . 1. Property of a rhombus. 2. AC ≅ AC . 2. 3. ∆ ACD ≅ ∆ CAB . 3. 4. ∠ CDA + ( ∠ DAC + ACD ) = 180 o . 4. 5. ∠ ACD ≅ ∠ CAB . 5. 6. 6. Substitution. 7. ∠ CDA & ∠ DAB are supplementary. 7. 8. 8. 9. Repeat 4 – 8 to show BC || DA . B Given: ABCD is a parallelogram. A 2 Prove: AC and BD bisect each other. 3 E
4 C 1
D Statement Justification
1. . 1. Alternate interior angles. 2. ∠ 3 ≅ ∠ 4 . 2. . 3. DA ≅ BC . 3. Opp. Sides of a parallelogram. 4. . 4. ASA. 5. AE ≅ EC . 5. . 6. . 6. CPCTC. 7. DB & AC bisect each other. 7. 5 & 6. B Given: ABCD is a parallelogram. A 2 Prove: AC and BD bisect each other. 3 E
4 C 1
D Statement Justification
1. ∠ 1 ≅ ∠ 2 . 1. Alternate interior angles. 2. ∠ 3 ≅ ∠ 4 . 2. Alternate interior angles. 3. DA ≅ BC . 3. Opp. Sides of a parallelogram. 4. ∆ DAE ≅ ∆ BCE . 4. ASA. 5. AE ≅ EC . 5. CPCTC. 6. DE ≅ EB . 6. CPCTC. 7. DB & AC bisect each other. 7. 5 & 6. A Given: ABCD is a parallelogram. B Prove: Opposite angles are congruent.
D C Statement Justification
ο 1. m ∠ A + m ∠ B = 180 . 1. . 2. . 2. Property of Co-interior angles. 3. m ∠ A + m ∠ B = m ∠ B + m ∠ C . 3. . 4. . 4. Cancellation. 5. . 5. Angles with equal measure are cong. 6. Similarly ∠ B ≅ ∠ D . 6. Steps 1-5. A Given: ABCD is a parallelogram. B Prove: Opposite angles are congruent.
D C Statement Justification
ο 1. m ∠ A + m ∠ B = 180 . 1. Property of Co-interior angles. ο 2. m ∠ B + m ∠ C = 180 . 2. Property of Co-interior angles. 3. m ∠ A + m ∠ B = m ∠ B + m ∠ C . 3. Substitution. 4. m ∠ A = m ∠ C . 4. Cancellation. 5. ∠ A ≅ ∠ C . 5. Angles with equal measure are cong. 6. Similarly ∠ B ≅ ∠ D . 6. Steps 1-5. A
Given: ABCD is a parallelogram & angle A is a right angle. B D Prove: ABCD has all right angles.
C Statement Justification
1. ∠ A ≅ ∠ C . 1. . ο 2. m ∠ A = 90 . 2. Property of a right angle. ο 3. m ∠ A + m ∠ B = 180 . 3. Property of Co-interior angles. 4. 90 ο + m ∠ B = 180 ο. 4. Substitution. 5. . 5. Subtraction. 6. ∠ B ≅ ∠ A . 6. Angles have equal measure. 7. . 7. Property of parallelograms. 8. ∠ A ≅ ∠ D . 8. . 9. All angles are right angles. 9. All angles are cong. to angle A. A
Given: ABCD is a parallelogram & angle A is a right angle. B D Prove: ABCD has all right angles.
C Statement Justification
1. ∠ A ≅ ∠ C . 1. Property of parallelograms. ο 2. m ∠ A = 90 . 2. Property of a right angle. ο 3. m ∠ A + m ∠ B = 180 . 3. Property of Co-interior angles. 4. 90 ο + m ∠ B = 180 ο. 4. Substitution. ο 5. m ∠ B = 90 . 5. Subtraction. 6. ∠ B ≅ ∠ A . 6. Angles have equal measure. 7. ∠ B ≅ ∠ D . 7. Property of parallelograms. 8. ∠ A ≅ ∠ D . 8. Transitive property. 9. All angles are right angles. 9. All angles are cong. to angle A. B
A Given: ABCD is a kite (diagram is to scale) . Prove: ∠ ABC ≅ ∠ ADC & AC bisects ∠DAB & ∠ BCD .
D C Statement Justification
1. AB ≅ DA & BC ≅ CD . 1. Property of a kite. 2. . 2. Reflexive axiom. 3. . 3. SSS. 4. . 4. 5. ∠ CAB ≅ ∠ CAD & ∠ BCA ≅ ∠ DCA . 5. 6. AC bisects ∠ DAB & ∠ BCD . 6. B
A Given: ABCD is a kite (diagram is to scale) . Prove: ∠ ABC ≅ ∠ ADC & AC bisects ∠DAB & ∠ BCD .
D C Statement Justification
1. AB ≅ DA & BC ≅ CD . 1. Property of a kite. 2. AC ≅ AC . 2. Reflexive axiom. 3. ∆ ABC ≅ ∆ ADC . 3. SSS. 4. ∠ ABC ≅ ∠ ADC . 4. CPCTC 5. ∠ CAB ≅ ∠ CAD & ∠ BCA ≅ ∠ DCA . 5. CPCTC 6. AC bisects ∠ DAB & ∠ BCD . 6. Definition of angle bisector. B
Given: ABCD is a kite (diagram is to scale) .A AC ⊥ BD AC BD Prove: & bisects . E
D C Statement Justification
1. & . 1. Property/definition of a kite. 2. . 2. 3. ∆ ABE ≅ ∆ ADE . 3. SAS. 4. . 4. CPCTC 5. . 5. ∠ BEA ≅ ∠ DEA & form a straight line. 6. . 6. 7. AC bisects BD . 7. Definition of segment bisector. B
Given: ABCD is a kite (diagram is to scale) .A AC ⊥ BD AC BD Prove: & bisects . E
D C Statement Justification
1. ∠ EAB ≅ ∠ EAD & AB ≅ AD . 1. Property of a kite. 2. AE ≅ AE . 2. Reflexive axiom. 3. ∆ ABE ≅ ∆ ADE . 3. SAS. 4. ∠ BEA ≅ ∠ DEA . 4. CPCTC 5. AC ⊥ BD . 5. ∠ BEA ≅ ∠ DEA & form a straight line. 6. BE ≅ DE . 6. CPCTC 7. AC bisects BD . 7. Definition of segment bisector. B Given: ABCD is a rhombus.
Prove: Diagonals are perpendicular A C bisectors.
D
Statement Justification
1. Diagonals are perpendicular. 1. A rhombus is a kite . 2. Diagonals are bisectors . 2. A rhombus is a parallelogram . 3. Diagonals are perpendicular bisectors. 3. Definition of perpendicular bisector. B Given: ABCD is a rhombus.
Prove: Diagonals are perpendicular A C bisectors.
D
Statement Justification
1. Diagonals are perpendicular. 1. A rhombus is a kite. 2. Diagonals are bisectors. 2. A rhombus is a parallelogram. 3. Diagonals are perpendicular bisectors. 3. Definition of perpendicular bisector. A
Given: ABCD is a rectangle. B Prove: All interior angles are right angles. D
C Statement Justification
1. One interior angle is a right angle . 1. Definition of a rectangle. 2. All interior angles are right angles . 2. A rectangle is a parallelogram . A
Given: ABCD is a rectangle. B Prove: All interior angles are right angles. D
C Statement Justification
1. One interior angle is a right angle. 1. Definition of a rectangle. 2. All interior angles are right angles. 2. A rectangle is a parallelogram & a parallelogram with one right angle has all right angles.