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Inequality in Triangle

In this section, we shall discuss inequality in triangle.

Theorem -1 : If two sides of a triangle are unequal, the longer side has greater angle opposite to it.

Given : A ΔABC in which AC > AB.

Prove that : ∠ABC > ∠ACB

Construction : Mark a point D on AC such that AB = AD. Join BD.

1) AB = AD 1) By Construction
2) ∠ ABD = ∠ADB 2) If two sides are equal then angle opposite to them are also equal
3)∠ADB > ∠DCB 3)As ∠ADB is an exterior angle of ΔBCD and exterior angle is always greater than interior angle.
4)∠ADB > ∠ACB 4)∠ACB = ∠DCB
5) ∠ABD > ∠ABC 5) ∠ABC = ∠ABD + ∠DBC
6) ∠ABC > ∠ACB 6) From (4) and (5)

Converse of the above theorem is also true.

Theorem : 2 In a triangle the greater angle has the longer side opposite to it.

1) In ΔABC, AC = 5cm, AB = 7 cm and BC = 3Cm. Write the angles in ascending order.
2)In ΔPQR, PQ = 8cm, PR = 3 cm and PQ= 6Cm. Write the angles in ascending order.
3) In ΔABC, AC is the longest side then which angle is the largest ?
4) In ΔPQR, QR is the shortes side then which angle is the smallest ?
5) In a right triangle MNO, right angled at N, which side is the longest side?
6) n ΔPQR, ∠P =40
0 ,∠Q =80 0 and ∠R =60 0 . Write the sides in ascending order.

Introduction to Triangles
Types of Triangles on the basis of Sides
Types of Triangles on the basis of Angles
Angle Sum Property of Triangles
Exterior and Interior angles of Triangle
Triangle Inequality Property
Congruent Triangles
Postulates of Congruent Triangle
Inequality in Triangle

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