#### Solved Examples and Worksheet for Triangle Inequality Theorem

Q1Sides PQ and PR are produced and SQR > TRQ. What is the relationship between PQ and PR?

A. PQ > PR
B. PQ < PR
C. PQ = 2PR
D. PQ = PR

Step: 1
SQR = 180o - (PQR)
[SQP is a straight angle.]
Step: 2
TRQ = 180o - (PRQ)
[TRP is a straight angle.]
Step: 3
180o - (PQR) > 180o - (PRQ)
[SQR > TRQ.]
Step: 4
PRQ > PQR
[Simplify.]
Step: 5
PQ > PR
[In a triangle the side opposite to the greater angle is greater.]
Correct Answer is :    PQ > PR
Q2Arrange the angles of ΔABC in descending order.

A. mC > mA > mB
B. mA > mB > mC
C. mB > mA > mC
D. mC > mB > mA

Step: 1
In a scalene triangle, the angle opposite to the larger side is larger.
Step: 2
As 14 > 11 > 8, we have mC > mB > mA
Correct Answer is :   mC > mB > mA
Q3The side lengths of a triangle are 5 cm, 7 cm and z cm respectively. Which of the following is true?

A. 2 < z < 12
B. 5 < z < 12
C. 5 < z < 7
D. 7 < z < 12

Step: 1
z < 5 + 7
[Triangle Inequality Theorem.]
Step: 2
z < 12
[In a triangle the difference of any two sides is less than the third side.]
Step: 3
z > 7 - 5
Step: 4
z > 2
Step: 5
2 < z < 12
[Step 2 and step 4.]
Correct Answer is :   2 < z < 12
Q4Which of the following can be the length of BC?

A. 9 cm
B. 5 cm
C. 2 cm
D. 23 cm

Step: 1
BC < 14 + 9
[Triangle Inequality Theorem.]
Step: 2
BC < 23 cm
Step: 3
BC > 14 - 9
[In a triangle, the difference of the lengths of any two sides is less than the third side.]
Step: 4
BC > 5 cm
Step: 5
5 cm < BC < 23 cm
Step: 6
As 9 lies between 5 and 23, BC can be 9 cm.
Correct Answer is :   9 cm
Q5Which of the following can be the length of BC?

A. 11 cm
B. 14 cm
C. 23 cm
D. 6 cm

Step: 1
BC < 17 + 6
[Triangle Inequality Theorem.]
Step: 2
BC < 23 cm
Step: 3
BC > 17 - 6
[In a triangle, the difference of the lengths of any two sides is less than the third side.]
Step: 4
BC > 11 cm
Step: 5
11 cm < BC < 23 cm
Step: 6
From the values given in the options, 14 cm lies between 11 cm and 23 cm.
Step: 7
So, the length of BC is 14 cm.
Correct Answer is :   14 cm
Q6AB > BC > CA. What could be the measures of A, B and C respectively?

A. 30, 40, 110
B. 30, 110, 40
C. 40, 30, 110
D. 110, 40, 30

Step: 1
AB > BC > CA C > A > B
[In a triangle, the greater angle is opposite to the greater side.]
Step: 2
Option (C) satisfies the above condition as for mA = 40, mB = 30 and mC = 110, C > A > B.
Correct Answer is :   40, 30, 110
Q7In a triangle, the smallest angle is 40o. The largest angle is

A. 139o
B. 99o
C. less than 100o
D. greater than 100o

Step: 1
Smallest angle + medium angle + largest angle = 180o
[Triangle-Angle-Sum theorem.]
Step: 2
40o + (Medium angle) + (Largest angle) = 180o
[Substitute.]
Step: 3
Medium angle = 140 - (largest angle)
[Simplify.]
Step: 4
Medium angle > smallest angle
Step: 5
140 - (largest angle) > 40o
[Step 3.]
Step: 6
Largest angle < 100o
[Simplify.]
Correct Answer is :    less than 100o
Q8In ΔABC, p cm and h cm are the perimeter and the sum of the altitudes of the triangle. Then what is the relationship between p and h?
A. p = 2h
B. p < h
C. p > h
D. p = h

Step: 1

Step: 2
[In a right triangle, hypotenuse is the longest side.]
Step: 3
Similarly, BC > BE and CA > CF
Step: 4
AB + BC + CA > AD + BE + CF
Step: 5
p > h
Correct Answer is :    p > h
Q9In ΔABC, let p cm and r cm denote the perimeter and the sum of its medians respectively. What is the relationship between p and r?
A. p > r
B. p = 2r
C. p < r
D. p = r

Step: 1
Let a, b, c be the length of the sides and m1, m2, m3 be the length of the medians.
Step: 2
[Produce AD to E such that AD = DE and join C to E.]

Step: 3
ΔBDA ΔCDE
[SAS postulate.]
Step: 4
CE = BA = c
[Step 3.]
Step: 5
In ΔAEC, AC + CE > AE
[Triangle Inequality Theorem.]
Step: 6
b + c > 2m1
[Substitute.]
Step: 7
Similarly, for other medians, we get c + a > 2m2 and a + b > 2m3
Step: 8
(b + c) + (c + a) + (a + b) > 2m1 + 2m2 + 2m3
Step: 9
2(a + b + c) > 2(m1 + m2 + m3)
Step: 10
(a + b + c) > m1 + m2 + m3
[Simplify.]
Step: 11
p > r
Correct Answer is :    p > r
Q10In ΔABC, A = 68o, B = 52o. Which is the greatest side?
A. AB
B. BC
C. CA
D. cannot be determined with the given data

Step: 1
In ΔABC, C = 180o - (A + B)
[Triangle Angle Sum theorem.]
Step: 2
C = 180o - (68o + 52o)
[Substitute.]
Step: 3
C = 60o
[Substitute.]
Step: 4

Step: 5
A is the greatest angle BC is the greatest side.
[In a triangle, the greatest side is opposite to the greatest angle.]
Q11In a triangle, which is the side opposite to the obtuse angle?
A. the smallest
B. neither the greatest nor the smallest
C. either the greatest or the smallest
D. the greatest

Step: 1
An obtuse angle is the greatest angle in the triangle.
[Triangle-Angle-Sum theorem.]
Step: 2
Side opposite to the greatest angle i.e., the obtuse angle is the greatest.
Correct Answer is :   the greatest
Q12In ΔPQR, PQ = PR, Q = 65o. Find the relationship between QR and PR.
A. QR = PR
B. QR = 2PR
C. QR < PR
D. QR > PR

Step: 1
R = Q = 65o
[As PQ = PR.]
Step: 2
P = 180o - (R + Q)
[Triangle Angle Sum theorem.]
Step: 3
P = 180o - (65o + 65o)
[Substitute.]
Step: 4
P = 50o
[Simplify.]
Step: 5

Step: 6
PR > QR
[In a triangle, the side opposite to the greater angle is greater.]
Step: 7
QR < PR
Correct Answer is :    QR < PR
Q13In ΔABC, mA = 42, mB = 78. What is the relationship between AC and AB?
A. AC < AB
B. AC = AB
C. AC > AB
D. AC = 2AB

Step: 1
C = 180o - (A + B)
[Triangle-Angle-Sum theorem.]
Step: 2
C = 180o - (42o + 78o)
[Substitute.]
Step: 3
C = 60o
[Simplify.]
Step: 4

Step: 5
B > C AC > AB
[In a triangle, the greater side is opposite to the greater angle.]
Correct Answer is :    AC > AB
Q14In ΔXYZ, X = 42o and Y = 52o. The bisectors of X and Y meet at O. Which of the following is true?

A. OY = 2OX
B. OY = OX
C. OY < OX
D. OY > OX

Step: 1

Step: 2
In ΔXOY, OYX > OXY OX > OY
[In a triangle the greater side is opposite to the greater angle.]
Step: 3
OY < OX.
Correct Answer is :    OY < OX
Q15In the figure, x is a whole number. What is the smallest possible value for x?

A. 1 unit
B. 7 units
C. 6 units
D. 12 units

Step: 1
2x > 13
[Sum of two sides shall be greater than the third side.]
Step: 2
Minimum value of x shall be 7 units.
[x is to be a whole number.]
Correct Answer is :   7 units
• Triangle