Inequalities in a Triangle with Examples - Triangles, Class 9, Mathematics

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

INEQUALITIES IN A TRIANGLE

THEOREM-5 : 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.

THEOREM-6 (Converse of Theorem-5) : 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 :(i) AC = AB(ii) AC < AB(iii) AC > ABOut of these exactly one must be true. AC = AB.∠s opposite to equal sides of a Δ are Case-I. ∠ABC = ∠ACBThis is contrary to what is given.∴ AC = AB is not true. equal Case-II. AC < AB.⇒    ∠ AB > AC⇒   ∠ACB > ∠ABCThis is contrary to what is given.∴     AC < AB is not true.Thus, we are left with the only possibilityAC > AB, which must be true. Greater side has greater angle opp. to it.

Hence, proved.

THEOREM-7 : 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⇒ ∠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 > BCor AB + AC > BC. Greater angle has greater side opp. to it.BAD is a straight line, BD = BA + AD. 4. Similarly, AB + BC > ACand BC + AC > AB. 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°⇒  ∠A + 40° = 100° Ext. ∠ = sum of int. opt. ∠sLinear 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° 5. ∠A > ∠B⇒ BC > AC 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° 4. ∠C > ∠A⇒ AB > BC . Greater angle has greater side opp. to it∠A = 60° and ∠B = 40° 5. ∠A > ∠B⇒ BC > AC Greater angle has greater side opp.to it.

 THINGS TO REMEMBER1. 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.
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## FAQs on Inequalities in a Triangle with Examples - Triangles, Class 9, Mathematics - Extra Documents & Tests for Class 9

 1. What are inequalities in a triangle?
Ans. Inequalities in a triangle refer to the relationships between the sides and angles of a triangle. There are three inequalities in a triangle: the triangle inequality, the exterior angle inequality, and the angle-side inequality.
 2. What is the triangle inequality?
Ans. The triangle inequality states that the sum of any two sides of a triangle must be greater than the third side. In other words, if a, b, and c are the sides of a triangle, then a + b > c, a + c > b, and b + c > a.
 3. What is the exterior angle inequality?
Ans. The exterior angle inequality states that the measure of an exterior angle of a triangle is greater than the measure of either of the remote interior angles. In other words, if A, B, and C are the interior angles of a triangle, and X is the measure of an exterior angle at vertex C, then X > A and X > B.
 4. What is the angle-side inequality?
Ans. The angle-side inequality states that the length of a side of a triangle is greater than the length of the side opposite a smaller angle. In other words, if a, b, and c are the sides of a triangle, and A, B, and C are the opposite angles, then a > b if A < B, b > c if B < C, and c > a if C < A.
 5. How can inequalities in a triangle be useful?
Ans. Inequalities in a triangle can be useful in a variety of ways. They can help determine whether a given set of side lengths and angles can form a triangle, and they can be used to prove other geometric relationships. They can also be used in real-world applications, such as calculating the maximum height of a ladder that can be placed against a wall.

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