Question

aa criteria Triangle theorem​

Answer 1

Question:-

If two angles of one triangle are respectively equal to two angles of another triangle. Then the Two triangles are similar.

AnswEr:-

Given :- Two triangles ∆ABC and ∆DEF

Such that, ∠B = ∠E & ∠C = ∠F

To prove:-

∆ABC ~ ∆DEF

$\rule{150}3$

To Proof :-

In ∆ABC

∠A + ∠B + ∠C = 180° [ By Angle Sum Property]____eq(1)

Similarly,

In ∆ DEF

∠D + ∠E + ∠F = 180° [By Angle Sum Property] ____eq(2)

From eqn (1) & (2)

↪ ∠A + ∠B + ∠C = ∠D + ∠E + ∠F

↪ ∠A + ∠E + ∠F = ∠D + ∠E + ∠F ( as ∠B = ∠E & ∠C = ∠F)

↪ ∠A = ∠D _____eq (3)

Thus, In ∆ ABC & ∆DEF

↪ ∠A = ∠D [From (3)]

↪∠B = ∠ E [Given]

↪∠C = ∠F [Given]

∴ ∆ ABC ~ ∆DEF [AAA similarity Criteria]

Hence Proved!

$\rule{150}3$

Answer 2

Question:-

If two angles of one triangle are respectively equal to two angles of another triangle. Then the Two triangles are similar.

AnswEr:-

Given :- Two triangles ∆ABC and ∆DEF

Such that, ∠B = ∠E & ∠C = ∠F

To prove:-

∆ABC ~ ∆DEF

To Proof :-

In ∆ABC

∠A + ∠B + ∠C = 180° [ By Angle Sum Property]____eq(1)

Similarly,

In ∆ DEF

∠D + ∠E + ∠F = 180° [By Angle Sum Property] ____eq(2)

From eqn (1) & (2)

↪ ∠A + ∠B + ∠C = ∠D + ∠E + ∠F

↪ ∠A + ∠E + ∠F = ∠D + ∠E + ∠F ( as ∠B = ∠E & ∠C = ∠F)

↪ ∠A = ∠D _____eq (3)

Thus, In ∆ ABC & ∆DEF

↪ ∠A = ∠D [From (3)]

↪∠B = ∠ E [Given]

↪∠C = ∠F [Given]

∴ ∆ ABC ~ ∆DEF [AAA similarity Criteria]

Hence Proved!

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