# Triangle Inequality Worksheet

Triangle Inequality Worksheet
• Page 1
1.
In the figure, $x$ is a whole number. What is the smallest possible value for $x$?

 a. 1 unit b. 6 units c. 7 units d. 12 units

#### Solution:

2x > 13
[Sum of two sides shall be greater than the third side.]

Minimum value of x shall be 7 units.
[x is to be a whole number.]

2.
Select the measurements that match the sides of a triangle.
 a. 7, 9, 19 b. 7, 12, 1 c. 7, 7, 7 d. 7, 7, 16

#### Solution:

The sum of the measures of any two sides of any triangle is greater than measure of the third side.
[Triangle inequality theorem.]

A triangle can have side lengths of 7, 7 and 7 as 7 + 7 > 7.
[Step 1.]

Other choices do not match the criteria that sum of any two sides shall be greater than the third side.
[Step 1.]

3.
Which of the following does not represent the lengths of the sides of a triangle?
 a. 2 cm, 6 cm, 7 cm b. 5 cm, 2 cm, 5 cm c. 5 cm, 5 cm, 8 cm d. 3 cm, 10 cm, 15 cm

#### Solution:

Sum of any two sides of a triangle shall be greater than the third side.

Out of the given choices, lengths of sides 3 cm, 10 cm, and 15 cm do not meet this.

So, these measurements do not represent the measurements of a triangle.

4.
Which cannot be the third side of a triangle which has two sides as 8 cm and 12 cm?
 a. 13 cm b. 22 cm c. 10 cm d. 5 cm

#### Solution:

Sum of any two sides of a triangle must be greater than the third side.
[Triangle Inequality Theorem.]

As 8 + 12 < 22, it is not possible to have a triangle with side lengths of 8, 12, and 22.

5.
The difference of any two sides of a triangle is
 a. equal to the third side b. is not related to the third side c. greater than the third side d. less than the third side

#### Solution:

a + b > c; b + c > a; c + a > b
[Triangle Inequality Theorem.]

c - b < a; a - c < b; b - a < c

The difference of any two sides of a triangle is less than the third side.

6.
Select the correct statement(s) with respect to a triangle.
I. Sides containing the smallest angle will be larger than the third side.
II. Sides containing the largest angle will be longer than the third side.
III. Sum of the lengths of the smaller sides will be less than the length of the larger side.
 a. III only b. I only c. II only d. I, II, and III

#### Solution:

Side opposite to the smallest angle will be the smallest.

So, sides containing the smallest angles will be longer than the third side, i.e., side opposite to the smallest angle.

Side opposite to the largest angle will be the longest.

So, sides containing the largest angle will not be longer than the third side.

Sum of the lengths of any two sides will be greater than the third side.

So, only statement I is correct.

7.
What is the ascending order for the lengths of the sides of ΔABC?

 a. AC < BC < AB b. AC < AB < BC c. AB < BC < AC d. BC < AB < AC

#### Solution:

mB = 180 Ã¢â‚¬â€œ (mA + mC)
[Triangle-Angle-Sum theorem.]

mB = 180 Ã¢â‚¬â€œ (40 + 60)
[Substitute.]

mB = 80
[Simplify.]

BC is the shortest side as it is opposite to A.
[40o < 60o < 80o.]

AC is the longest side as it is opposite to B.
[40o < 60o < 80o.]

So, the ascending order is, BC < AB < AC.

8.
Select the correct statement/statements.
1. BC is always greater than 8 cm.
2. BC is always equal to 8 cm.
3. BC is always greater than 2 cm.
4. BC is always less than 8 cm.

 a. 1 and 2 only b. 3 only c. 1 only d. 3 and 4 only

#### Solution:

AB = 3, AC = 5, BC < 3 + 5
[Triangle Inequality Theorem.]

BC < 8 cm

BC > 5 - 3

BC > 2

So, statements 3 and 4 only are correct.

9.
If $\angle$A is the largest angle in ΔABC, then which of the following can be the length of BC?

 a. 2 cm b. 9 cm c. 6 cm d. 4 cm

#### Solution:

A is the largest angle.

The longest side is BC.

But 8 cm > BC > 5 cm

Since AC = 5 cm, BC > 5 cm.

The appropriate choice for BC is 6 cm.

10.
Sides PQ and PR are produced and $\angle$SQR > $\angle$TRQ. What is the relationship between PQ and PR?

 a. PQ > PR b. PQ = PR c. PQ = 2PR d. PQ < PR

#### Solution:

SQR = 180o - (PQR)
[SQP is a straight angle.]

TRQ = 180o - (PRQ)
[TRP is a straight angle.]

180o - (PQR) > 180o - (PRQ)
[SQR > TRQ.]

PRQ > PQR
[Simplify.]

PQ > PR
[In a triangle the side opposite to the greater angle is greater.]