The image contains four geometric proofs involving congruence of triangles and quadrilaterals. By applying various postulates of congruence (SSS, AA, ASA, SAS), we can prove the congruence based on the given conditions.
Sure, let's solve these geometric proofs.
Proof 6: The given conditions are BC = CD and AC = CE. We aim to prove that ΔACB ≅ ΔECD. By the Side-Side-Side (SSS) postulate of congruence, if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. Here, we have two pairs of congruent sides (BC = CD and AC = CE). If we can prove that AB = DE, then by the SSS postulate, ΔACB would be congruent to ΔECD.
Proof 7: The given conditions are ∠Q = ∠S and ∠QTR = ∠SRT. We aim to prove that ΔQTR ≅ ΔSRT. By the Angle-Angle (AA) postulate of congruence, if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Here, we have two pairs of congruent angles (∠Q = ∠S and ∠QTR = ∠SRT). If we can prove that QR = SR, then by the Side-Angle-Side (SAS) postulate, ΔQTR would be congruent to ΔSRT.
Proof 8: The given conditions are ∠L = ∠K and ∠L = ∠KL. We aim to prove that ΔHIL ≅ ΔJKL. By the Angle-Side-Angle (ASA) postulate of congruence, if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. Here, we have two pairs of congruent angles (∠L = ∠K and ∠L = ∠KL). If we can prove that HI = JK, then by the ASA postulate, ΔHIL would be congruent to ΔJKL.
Proof 9: The given conditions are AB = CD and ∠ADB = ∠CDB. We aim to prove that quadrilateral ADAB is congruent to ABCD. By the Side-Angle-Side (SAS) postulate of congruence for quadrilaterals, if two sides and the included angle of one quadrilateral are congruent to two sides and the included angle of another quadrilateral, then the quadrilaterals are congruent. Here, we have a pair of congruent sides (AB = CD) and a pair of congruent angles (∠ADB = ∠CDB). If we can prove that AD = BC, then by the SAS postulate, quadrilateral ADAB would be congruent to ABCD.