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Answers
Explanation:
We have to prove that the difference of two sides of a triangle is less than the third side.
Refer to the attached image.
To prove: AC-AB< BC
Construction: Mark a point 'D' on AC such that the distance AB=AD
Therefore, we have to prove that AC-AD<BC
So, we have to prove that CD < BC
Proof:
Consider triangle ABD,
since AB=AD
Since, opposite angles opposite to the equal opposite sides are always equal.
So, \angle ABD=\angle ADB∠ABD=∠ADB
So, p = p
Now, let \angle DBC=x , \angle BDC=y∠DBC=x,∠BDC=y
Now, using exterior angle property in triangle ABD,
Exterior angle property of a triangle states that the measure of an exterior angle is equal to the sum of the two interior angles.
y=p+\angle Ay=p+∠A (equation 1)
Now, using exterior angle property in triangle BCD,
p=x+\angle Cp=x+∠C
x=p-\angle Cx=p−∠C (equation 2)
Now, by comparing equation 1 and 2,
p+\angle A>p-\angle Cp+∠A>p−∠C
So, y > x
BC > CD
CD < BC
AC - AD < BC
Since, AD = AB
Therefore, AC - AB < BC
Hence, proved.
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