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Problem 1: In triangles DEA and BCE, DE ≅ EC (given), AE ≅ EB (given), and ∠DEA ≅ ∠BEC (vertical angles). By SAS (Side-Angle-Side) congruence, ΔDEA ≅ ΔBCE.
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Problem 2: Since GH || JI and GH ≅ JI, quadrilateral GHIJ is a parallelogram. Thus, GJ ≅ HI and HJ ≅ JG (opposite sides of a parallelogram are congruent). Also, HJ is common to both triangles. By SSS (Side-Side-Side) congruence, ΔGJH ≅ ΔHJI.
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Problem 3: Given ∠XYP ≅ ∠XNP and ∠WNX ≅ ∠YW X, and since WY and WX are common sides, by AAS (Angle-Angle-Side) congruence, ΔWNX ≅ ΔWYX.
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Problem 4: BD bisects ∠B, so ∠CBD ≅ ∠EBD. BD ⊥ CE implies ∠BDC ≅ ∠BDE (both right angles). BD is common to both triangles. By ASA (Angle-Side-Angle) congruence, ΔBDC ≅ ΔBDE.
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Problem 5: BD ⊥ AB and BD ⊥ DE imply ∠ABD ≅ ∠EDB (both right angles). BC ≅ DC (given). BD is common. By SAS (Side-Angle-Side) congruence, ΔABD ≅ ΔEDB. Therefore, ∠A ≅ ∠E (corresponding parts of congruent triangles).
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Problem 6: Z is the midpoint of YP, so YZ ≅ PZ. Given ∠Y ≅ ∠P, and ∠YZX ≅ ∠PZL (vertical angles). By ASA (Angle-Side-Angle) congruence, ΔXYZ ≅ ΔPLZ.
Parent Tip: Review the logic above to help your child master the concept of triangle proofs worksheet with answers.
