Question
Given: Rectangle ABCD
Prove: AC¯¯¯¯¯≅BD¯¯¯¯¯
Statement Reason
______________ Opposite sides of a rectangle are congruent.
DC¯¯¯¯¯≅DC¯¯¯¯¯ ______________
∠ADC and ∠BCD are right angles. Definition of rectangle
___________________ All right angles are congruent.
△ADC≅△BCD ___________________
AC¯¯¯¯¯≅BD¯¯¯¯¯ _______________
1. AD¯¯¯¯¯≅BC¯¯¯¯¯
2. AB¯¯¯¯¯≅DC¯¯¯¯¯
3. AC¯¯¯¯¯≅ BD¯¯¯¯¯
4. △ACD≅△BDC
5. △ADC≅△BCD
6. Reflective Property Congruence
7. Transitive Property of Congruence
8.CPCTC
9. HL Congruence Postulate
10. SAS Congruence Postulate
Prove: AC¯¯¯¯¯≅BD¯¯¯¯¯
Statement Reason
______________ Opposite sides of a rectangle are congruent.
DC¯¯¯¯¯≅DC¯¯¯¯¯ ______________
∠ADC and ∠BCD are right angles. Definition of rectangle
___________________ All right angles are congruent.
△ADC≅△BCD ___________________
AC¯¯¯¯¯≅BD¯¯¯¯¯ _______________
1. AD¯¯¯¯¯≅BC¯¯¯¯¯
2. AB¯¯¯¯¯≅DC¯¯¯¯¯
3. AC¯¯¯¯¯≅ BD¯¯¯¯¯
4. △ACD≅△BDC
5. △ADC≅△BCD
6. Reflective Property Congruence
7. Transitive Property of Congruence
8.CPCTC
9. HL Congruence Postulate
10. SAS Congruence Postulate
Answers
GPT-5 mini
Fill in the missing statements/reasons as follows:
Statement Reason
1. AD ≅ BC Opposite sides of a rectangle are congruent.
2. DC ≅ DC Reflective Property (congruence)
3. ∠ADC and ∠BCD are right angles. Definition of rectangle
4. ∠ADC ≅ ∠BCD All right angles are congruent.
5. △ADC ≅ △BCD SAS Congruence Postulate
6. AC ≅ BD CPCTC
Mapping to the numbered choices you gave:
- AD ≅ BC = choice 1
- DC ≅ DC reason = choice 6 (Reflective Property)
- △ADC ≅ △BCD reason = choice 10 (SAS)
- AC ≅ BD reason = choice 8 (CPCTC)
Reasoning: From the rectangle we get AD ≅ BC and ∠ADC, ∠BCD are right angles. With DC common to both triangles and right angles congruent, triangles ADC and BCD are congruent by SAS, so corresponding diagonals AC and BD are congruent by CPCTC.
Statement Reason
1. AD ≅ BC Opposite sides of a rectangle are congruent.
2. DC ≅ DC Reflective Property (congruence)
3. ∠ADC and ∠BCD are right angles. Definition of rectangle
4. ∠ADC ≅ ∠BCD All right angles are congruent.
5. △ADC ≅ △BCD SAS Congruence Postulate
6. AC ≅ BD CPCTC
Mapping to the numbered choices you gave:
- AD ≅ BC = choice 1
- DC ≅ DC reason = choice 6 (Reflective Property)
- △ADC ≅ △BCD reason = choice 10 (SAS)
- AC ≅ BD reason = choice 8 (CPCTC)
Reasoning: From the rectangle we get AD ≅ BC and ∠ADC, ∠BCD are right angles. With DC common to both triangles and right angles congruent, triangles ADC and BCD are congruent by SAS, so corresponding diagonals AC and BD are congruent by CPCTC.