INEQUALITIES IN A TRIANGLE
THEOREM5 : If two sides of a triangle are unequal, then the greater side has greater angle opposite to it.
Given : A ΔABC in which AC > AB.
To prove : ∠ABC > ∠ACB.
Construction : Mark a point D on AC such that AD = AB. Join BD. D
Proof :
STATEMENT  REASON  
1.  AB = AD  By construction 
2.  ∠ABD = ∠BDA  ∠s opposite to equal sides of a Δ are equal 
3.  ∠BDA > ∠DCB  (Ext. ∠ of ΔBCD) > (Each of its int. Opp. ∠s) 
4.  ∠ABD > ∠DCB  Using (2) 
5.  ∠ABC > ∠ABD  ∠ABD is a part of ∠ABC. 
6.  ∠ABC > ∠DCB  Using (5) 
7.  ∠ABC > ∠ACB  ∠DCB = ∠ACB. 
THEOREM6 (Converse of Theorem5) : If two angles of a triangle are unequal, then the greater
angle has greater side opposite to it.
Given : A ΔABC in which ∠ABC > ∠ACB.
To prove : AC > AB.
Proof :
STATEMENT  REASON  
 We may have three possibilities only :  AC = AB. 
CaseI. ∠ABC = ∠ACB  equal  
CaseII. AC < AB. 
Greater side has greater angle opp. to it. 
Hence, proved.
THEOREM7 : The sum of any two sides of a triangle is greater than its third side.
Given : A ΔABC.
To prove : (i) AB + AC > BC
(ii) AB + BC > AC
(iii) BC + AC > AB.
Construction : Produce BA to D such that AD = AC. Join CD.
Proof :
 STATEMENT  REASON 
1.
 AD = AC  By construction ∠s opposite to equal sides of a Δ are equa 
2.  ∠BCD > ∠ACD  Using (1) & (2) 
3.
 ∠BCD > ∠ADC  Greater angle has greater side opp. to it. BAD is a straight line, BD = BA + AD. 
4.
 Similarly, AB + BC > AC  AD = AC, by construction 
REMARK :(i) The largest side of a triangle has the greatest angle opposite to it and converse is also true.
(ii) The smallest side of a triangle has the smallest angle opposite to it and converse in also true.
Ex.19 In fig, show that :
(i) AB > AC (ii) AB > BC and (iii) BC > AC.
Sol. Given : A ΔABC in which ∠B = 40° and ∠ACD = 100°.
To prove : (i) AB > AC
(ii) AB > BC D
(iii) BC > AC.
Proof :
STATEMENT  REASON  
1.
 ∠A + ∠B = 100°  Ext. ∠ = sum of int. opt. ∠s Linear pair of angles.

2.
 ⇒ ∠A = 60° ∠C + 100° = 180°  ∠C = 80° and ∠B = 40° 
3.
 ∠C > ∠B  ∠C = 80° and ∠A = 60° 
4.  ∠C > ∠A  ∠A = 60° and ∠B = 40° 
5.  ∠A > ∠B ⇒ BC > AC 
Greater angle has greater side opp. to it. 
STATEMENT  REASON  
1.
 ∠A + ∠B = 100°  Ext. ∠ = sum of int. opt. ∠s 
2.
 ⇒ ∠A = 60°  Linear pair of angles. ∠C = 80° and ∠B = 40° 
3.
 ⇒ ∠C = 80°  Greater angle has greater side opp. to it. ∠C = 80° and ∠A = 60° 
4.  ∠C > ∠A  Greater angle has greater side opp. to it ∠A = 60° and ∠B = 40° 
5.  ∠A > ∠B  Greater angle has greater side opp.to it. 
THINGS TO REMEMBER 1. Two figures are congruent, if they are of the same shape and of the same size. 2. Two circles of the same radii are congruent. 3. Two squares of the same sides are congruent. 4. If two triangles ABC and PQR are congruent under the correspondence A P, B Qand C R, then symbolically, it is expressed as Δ ABC ≌ Δ PQR. 5. If two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle, then the two triangles are congruent (SAS congruence rule). 6. If two angles and the included side of one triangle are equal to two angles and the included side of the other triangle, then the two triangles are congruent (ASA congruence rule). 7. If two angles and one side of one triangle are equal to two angles and the corresponding side of the other triangle, then the two triangles are congruent (AAS Congruence rule). 8. Angle opposite to equal sides of a triangle are equal. 9. Sides opposite to equal angles of a triangle are equal. 10. Each angle of an equilateral triangle is of 60°. 11. If three sides of one triangle are equal to three sides of the other triangle, then the two triangles are congruent (SSS congruence rule). 12. If in two right triangles, hypotenuse and one side of a triangle are equal to the hypotenuse and one side of other triangle, then the two triangles are congruent (RHS congruence rule). 13. In a triangle, angle opposite to the longer side is larger (greater). 14. In a triangle, side opposite to the larger (greater) angle is longer. 15. Sum of any two sides of a triangle is greater than the third side. 
1 videos228 docs21 tests

1. What are inequalities in a triangle? 
2. What is the triangle inequality? 
3. What is the exterior angle inequality? 
4. What is the angleside inequality? 
5. How can inequalities in a triangle be useful? 
1 videos228 docs21 tests


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