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Math՚s: Congruence: Angles Opposite to Equal Sides of a Triangle and Vice Versa
Angles Opposite to Equal Sides of a Triangle and Vice Versa
Based on the criteria for congruence of triangles, there are some important theorems related to the isosceles triangles which are summarized as follows-
Theorem 1- The angles opposite to the equal sides of a triangle are equal.
Given: A triangle ABC in which
To prove:
Construction: Draw bisector of meeting at .
Proof: In and ,
(Given)
(By construction)
and (Common)
(SAS)
Hence (c. p. c. t)
Hence the theorem.
Theorem 2- The sides opposite to the equal angles of a triangle are equal.
Given: A triangle in which
To Prove:
Construction: Draw bisector of meeting at .
Proof: In and ,
(Given)
(By construction)
and (Common)
(SAS)
Hence (c. p. c. t)
Hence the theorem.
Theorem 3- Perpendiculars (altitudes) drawn to equal sides, from opposite vertices of an isosceles triangle are equal.
Given: In and
To Prove:
Proof: In and
(Angles opposite equal sides of a triangle)
(Common)
∴ (AAS)
Hence, (c. p. c. t.)
Inequalities in a Triangle
There are some relations among sides and angles of a triangle when they are unequal. These are summarized as theorems
Theorem 1- If two sides of a triangle are unequal, then the longer side has the greater angle opposite to it.
Given: A triangle in which .
To Prove:
Construction: Make a point on the side such that and join .
Proof: In ,
∴ (Angles opposite equal sides)
But (Exterior angle of a triangle is greater than opposite
interior angle)
Again (Point D lies in the interior of the ∠ ACB) .
∴
Theorem 2- Sum of any two sides in a triangle is greater than the third side.
Given: is a triangle
To Prove: and
Construction: Produce to such that , Join and
Proof: … (1) Since,
… (2)
Therefore,
from (1) & (2)
In (Greater angle has greater side opposite to it)
()