1. In each pair of triangles given below, parts shown by identical marks are congruent. State the test and the one to one correspondence of vertices by which triangles in each pair are congruent and remaining congruent parts.
2*. In the adjacent figure, segment AD ≅ segment EC.
Which additional information is needed to show that ∆ABD and ∆EBC will be congruent by A-A-S (Angle Angle Side/Segment) test? {Figures Attached}
Answers
1. i. In ∆MST and ∆TBM,
∴ side MS ≅ side TB … [Given]
m∠MST = m∠TBM = 90° … [Given]
hypotenuse MT ≅ hypotenuse MT …[Common side]
∴ ∆MST ≅ ∆TBM …[by hypotenuse-side test]
∴ side ST ≅ side BM …[Corresponding sides of congruent triangles]
∠SMT ≅ ∠BTM …[Corresponding sides of congruent triangles]
∠STM ≅ ∠BMT …[Corresponding sides of congruent triangles]
ii. In ∆PRQ and ∆TRS, side PR ≅ side TR … [Given]
∠PRQ ≅ ∠TRS …[Vertically opposite angles]
side RQ ≅ side RS … [Given]
∴ ∆PRQ ≅ ∆TRS …[by SAS test]
∴ side PQ ≅ side TS …[Corresponding sides of congruent triangles]
∠RPQ ≅ ∠RTS …[Corresponding sides of congruent triangles]
∠PQR ≅ ∠TSR …[Corresponding sides of congruent triangles]
iii. In ∆DCH and ∆DCF,
∠DCH ≅ ∠DCF …[Given]
∠DHC ≅ ∠DFC …[Given]
side DC ≅ side DC …[Common side]
∴ ∆DCH ≅ ∆DCF …[by AAS test]
∴ side HC ≅ side FC …[Corresponding sides of congruent triangles]
side DH ≅ side DF…[Corresponding sides of congruent triangles]
∠HDC ≅ ∠FDC ….[Corresponding sides of congruent triangles]
2. Answer in the given attachment
1. In fig.(1)
MS=BT
∠MST=∠MBT=90 ⁰
MT=MT
So they are congruent by RHS criteria.
In fig.(2) They are congurent by SAS criterion.
In fig.(3) They are congurnt by RHS criterion.
2. We have given that AD = EC. Also we have angles ABD and EBC equal since they are vertically opposite. Hence if any of the other corresponding pair of angles are given equal then the given triangles will be congruent by A-A-S
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