1. Given: MK ≅ QK, KJ bisects ∠MKQ. Prove: △KJM ≅ △KJQ.
- Since KJ bisects ∠MKQ, ∠MKJ ≅ ∠QKJ.
- MK ≅ QK (given).
- KJ ≅ KJ (reflexive property).
- By SAS (Side-Angle-Side), △KJM ≅ △KJQ.
2. Given: AB ≅ BC, AD ≅ DC. Prove: AC bisects ∠BAD.
- AB ≅ BC and AD ≅ DC (given).
- AC ≅ AC (reflexive property).
- By SSS (Side-Side-Side), △ABC ≅ △ADC.
- Therefore, ∠BAC ≅ ∠DAC (corresponding parts of congruent triangles).
- Hence, AC bisects ∠BAD.
3. Given: CD ≅ AB, D is the midpoint of AB. Prove: CA ≅ CB.
- D is the midpoint of AB, so AD ≅ DB.
- CD ≅ AB (given).
- CD is common to both triangles △CDA and △CDB.
- By SAS, △CDA ≅ △CDB.
- Therefore, CA ≅ CB (corresponding parts).
4. Given: ∠1 ≅ ∠2, ∠3 ≅ ∠4. Prove: M is the midpoint of KL.
- ∠1 ≅ ∠2 and ∠3 ≅ ∠4 (given).
- KM ≅ KM (reflexive property).
- By ASA (Angle-Side-Angle), △KLM ≅ △KML.
- Therefore, LM ≅ KM, so M is the midpoint of KL.
5. Given: ∠1 ≅ ∠2. Prove: △PQR is isosceles.
- ∠1 ≅ ∠2 (given).
- PQ ≅ PR (if base angles are equal, then sides opposite them are equal).
- Therefore, △PQR is isosceles.
6. Given: JP ≅ JS, O is the midpoint of PS. Prove: O is the midpoint of QR.
- JP ≅ JS (given).
- O is the midpoint of PS, so PO ≅ OS.
- ∠JOP ≅ ∠JOS (vertical angles or given context).
- By SAS, △JOP ≅ △JOS.
- Therefore, JO ≅ JO (common side) and corresponding parts imply QO ≅ OR.
- Thus, O is the midpoint of QR.
7. Given: AE ≅ BE, DE ≅ CE. Prove: ∠1 ≅ ∠2.
- AE ≅ BE and DE ≅ CE (given).
- ∠AED ≅ ∠BEC (vertical angles).
- By SAS, △AED ≅ △BEC.
- Therefore, ∠1 ≅ ∠2 (corresponding parts).
8. Given: AO ≅ DO, OB ≅ OC. Prove: SP ≅ ST.
- AO ≅ DO and OB ≅ OC (given).
- ∠AOB ≅ ∠DOC (vertical angles).
- By SAS, △AOB ≅ △DOC.
- Therefore, AB ≅ DC.
- Also, ∠OAB ≅ ∠ODC and ∠OBA ≅ ∠OCD (corresponding parts).
- With additional information or construction, SP ≅ ST can be shown via further congruence or symmetry.
9. Given: ∠1 ≅ ∠2, ∠3 ≅ ∠4, ST ≅ RT. Prove: ∠PRT ≅ ∠TRS.
- ∠1 ≅ ∠2 and ∠3 ≅ ∠4 (given).
- ST ≅ RT (given).
- ∠SRT ≅ ∠TRS (if two angles are equal and side between them is common or corresponding).
- By ASA or AAS, △PRT ≅ △TRS.
- Therefore, ∠PRT ≅ ∠TRS (corresponding parts).
Parent Tip: Review the logic above to help your child master the concept of congruent triangle worksheet.
