Inequalities in a Triangle with Examples - Triangles, Class 9, Mathematics | Extra Documents & Tests for Class 9 PDF Download

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.

STATEMENT

REASON

We may have three possibilities only :
(i) AC = AB
(ii) AC < AB
(iii) AC > AB
Out of these exactly one must be true.

AC = AB.
∠s opposite to equal sides of a Δ are

Case-I. 

∠ABC = ∠ACB
This is contrary to what is given.
∴ AC = AB is not true.

equal

Case-II. AC < AB.
⇒    ∠ AB > AC
⇒   ∠ACB > ∠ABC
This is contrary to what is given.
∴     AC < AB is not true.
Thus, we are left with the only possibility
AC > AB, which must be true.

Greater side has greater angle opp. to it.

REASON

1.

AD = AC
⇒ ∠ACD = ∠ADC

By construction

∠s opposite to equal sides of a Δ are equa

2.

∠BCD > ∠ACD

Using (1) & (2)

3.

∠BCD > ∠ADC
⇒    BD > BC
⇒     BA + AD > BC
⇒     BA + AC > BC
or AB + AC > BC.

Greater angle has greater side opp. to it.

BAD is a straight line, BD = BA + AD.

4.

Similarly, AB + BC > AC
and BC + AC > AB.

STATEMENT

REASON

1.

∠A + ∠B = 100°
⇒  ∠A + 40° = 100°

Ext. ∠ = sum of int. opt. ∠s

Linear pair of angles.

2.

⇒   ∠A = 60° ∠C + 100° = 180°
⇒ ∠C = 80°

∠C = 80° and ∠B = 40°
Greater angle has greater side opp. to it.

3.

∠C > ∠B
⇒ AB > AC
 

∠C = 80° and ∠A = 60°
Greater angle has greater side opp. to it

4.

∠C > ∠A
⇒ AB > BC .

∠A = 60° and ∠B = 40°

Greater angle has greater side opp. to it.

STATEMENT

REASON

1.

∠A + ∠B = 100°
⇒  ∠A + 40° = 100°

Ext. ∠ = sum of int. opt. ∠s

2.

⇒   ∠A = 60°
∠C + 100° = 180°

Linear pair of angles.

∠C = 80° and ∠B = 40°

3.

⇒ ∠C = 80°
∠C > ∠B
⇒ AB > AC

Greater angle has greater side opp. to it.

∠C = 80° and ∠A = 60°

∠C > ∠A
⇒ AB > BC .

 Greater angle has greater side opp. to it

∠A = 60° and ∠B = 40°

∠A > ∠B
⇒ BC > AC

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.