PROPERTIES OF TRIANGLES

Find the values of the variables, give brief reasons.

Problem 1 :

Solution :

Property :

Sum of the interior angles = 180

m - 10 + m + m + 10 = 180

3m = 180

Divide by 3 on both sides.

m = 180/3

m = 60

Problem 2 :

Solution :

Property :

One of the sides of the triangle and the line passes through the other two sides are parallel, corresponding angles are equal.

a = 80

Sum of interior angles of a triangle = 180

50 + b + a = 180

50 + b + 80 = 180

130 + b= 180

b = 180 - 150

b = 30

Problem 3 :

Solution :

Property :

At the point of intersection, vertically opposite angles are equal.

x = 80

y + 40 + x = 180

y + 40 + 80 = 180

y + 120 = 180

Subtracting 120 on both sides.

y = 180 - 120

y = 60

Problem 4 :

Solution :

Property :

In any isosceles triangle, two of the angles will be equal. The sides which is opposite to the equal angle measure will be equal in length.

Since ∠B = ∠C

 x = 12 cm

Problem 5 :

Solution :

Property :

Using exterior angle theorem.

x = a + b

x = 62 + 80

x = 142

Problem 6 :

Triangle TAB has a perimeter of 40 cm. Could the measures of the sides, as shown, actually represent the measures of the sides of a triangle?

Solution :

Perimeter of the triangle TAB = 40 cm

4x + x + 3 + 2x + 2 = 40

7x + 5 = 40

Subtracting 5 on both sides.

7x = 40 - 5

7x = 35

Dividing by 7 on both sides.

x = 35/7

x = 5

So, the sides of the triangle are 12, 8 and 20

Using triangle inequality theorem :

12 + 8 > 20

8 + 20 > 12

12 + 20 > 8

So, these measures will not be the sides of the triangle.

Problem 7 :

Two sides of an isosceles triangle measure 4 and 8. Which of the following choices could be the measure of the third side?

(a)  10   (b)  8    (c) 6    (d)  4

Solution :

Let the sides of the triangle be a, b and c.

If a = 4, b = 8, then c = 4 or 8.

Case 1 :

Let a = 4, b = 8 and c = 4

4 + 8 > 4 (True)

4 + 4 > 8 (Not true)

Since these sides are not satisfying triangle inequality theorem, it will not be sides of the triangle.

Case 2 :

Let a = 4, b = 8 and c = 8

4 + 8 > 8 (True)

8 + 8 > 4 (True)

Since these sides are satisfying the triangle inequality theorem, it will be sides of the triangle.

So, the third side must be 8.

Problem 8 :

In ΔGAB, m∠G = 35º and m∠A = 52º. Which side of the triangle is the longest side?

Solution :

In triangle, the angles measures are m∠G,  m∠A,  m∠B.

m∠G + m∠A + m∠B = 180

35 + 52 + m∠B = 180

84 + m∠B = 180

Subtracting 84 on both sides.

m∠B = 180 - 84

m∠B = 96

Since the largest angle measure is m∠B, then the side which is opposite to this angle measure will be the longest side.

The longest side is AG.

Problem 9 :

In ΔABC, AB = 4x + 2, BC = 5x - 3, and AC = 2x + 4. The perimeter of the triangle is 36 units. Which angle in the triangle has the largest measure?

Solution :

Perimeter of the triangle ABC = 36

AB = 4x + 2, BC = 5x - 3, and AC = 2x + 4.

4x + 2 + 5x - 3 + 2x + 4 = 36

11x + 2 - 3 + 4 = 36

11x + 3 = 36

Subtracting 3 on both sides.

11x = 36 - 3

11x = 33

Dividing by 11 on both sides.

x = 33/11

x = 3

So, the largest measure is 14, that is AB. Angle C is the greatest measure.