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Home / Geometry / If two angles of triangles are equal, then the sides opposite to the equal angles are equal.

# If two angles of triangles are equal, then the sides opposite to the equal angles are equal.

Particular enunciation: Let ABC be a triangle in which the ∠ACB = the ∠ABC.

We have to prove that AB =AC.

Proof: If AC and AB are equal, suppose that AB>AC.

From BA cut off BD equal to CA, join CD.

Now in the ∆ABC and ∆DBC, AC=BD, BC is common and the included, ∠BCA = the included ∠DBC.

∴ ∆ABC ≌∆DBC [by SAS]

That is, area of ∆ABC = area of ∆DBC. But this is absurd, because a

Part cannot be equal to the whole.

∴ AB is not greater than AC. Similarly AC is not greater than AB.

∴ AB= AC (proved)

## If two triangles have the three sides of the one equal to the three sides of the other, each to each, then they are equal in all respects.

If two triangles have the three sides of the one equal to the three sides ...