Question

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prove that angles opposite to equal sides of an equilateral triangle are equal

Answer 1

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Answer 2

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theorem 7.2 : angles opposite to equal sides of an isoceles traingle are equal.

THIS RESULT CAN BE PROVED IN MANY WAYS . ONE OF THE PROOFS IS GIVEN HERE.

PROOF;- WE ARE GIVEN AN ISOCELES TRAINGLE ABC IN WHICH AB= AC . WE NEED TO PROVE THAT <B = <C

LET US DRAW THE BISECTOR OF <A AND LET D BE THE POINT OF INTERSECTION OF THIS BISECTOR OF <A AND BC( SEE THE FIGURE ATTACHED BELOW)

IN TRAINGLE BAD AND TRAINGLE CAD

AB= AC (GIVEN)

<BAD=<CAD (BY CONSTRUCTION)

AD=AD (COMMON)

TRAINGLE BAD≈ TRAINGLE CAD (BY SAS RULE )

SO, <ABD = <ACD, SINCE THEY ARE CORRESPONDING ANGLES OF CONGRUENT TRAINGLES

SO, <B=<C

IS THE CONVERSE ALSO TRUE ? THAT IS;

IF TWO ANGLES OF ANY TRAINGLE ARE EQUAL , CAN WE CONCLUDE THAT THE SIDES OPPOSITE TO THEM ARE ALSO EQUAL?

PERFORM THE FOLLOWING ACTIVITY.

CONSTRUCT A TRAINGLE ABC AND BC OF ANY LENGTH AND <B= <C=50°. DRAW THE BISECTOR OF <A AND LET IT INTERSECT BC AT D SEE THE SECOND ATTATCHMENT

CUT OUT THE TRAINGLE FROM THE SHEET OF PAPER AND FOLD IT ALONG AD SO THAT VERTEX C FALLS ON VERTEX B

OBSERVE THA AC COVERS AB COMPLETLY

SO, AC = AB

REPEAT THIS ACTIVITY WITH SOME MORE TRAINGLES EACH TIME YOU WILL OBSERVE THAT THE SIDES OPPOSITE TO EQUAL ANGLES ARE EQUAL . SO WE HAVE THE FOLLOWING

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