Triangles — Exercise 15

In each of the following, find the marked unknown angles :

Answer

From the figure, the three angles of the triangle are 70°, 72° and z.

By the angle sum property of a triangle,

⇒ 70° + 72° + z = 180°

⇒ 142° + z = 180°

⇒ z = 180° − 142°

⇒ z = 38°

Hence, z = 38°.

In each of the following, find the marked unknown angles :

Answer

From the figure, the upper triangle has angles 80°, 50° and b.

By the angle sum property of a triangle,

⇒ 80° + 50° + b = 180°

⇒ 130° + b = 180°

⇒ b = 180° − 130°

⇒ b = 50°

In the lower triangle, the angles are 40°, 45° and a.

By the angle sum property of a triangle,

⇒ 40° + 45° + a = 180°

⇒ 85° + a = 180°

⇒ a = 180° − 85°

⇒ a = 95°

Hence, a = 95° and b = 50°.

In each of the following, find the marked unknown angles :

Answer

From the figure,

The angle at the top vertex is made up of 20° and 45°, so the top angle is (20° + 45°) = 65°. The other two angles of the triangle are x and 60°.

By the angle sum property of a triangle,

⇒ 65° + x + 60° = 180°

⇒ 125° + x = 180°

⇒ x = 180° − 125°

⇒ x = 55°

Hence, x = 55°.

Can a triangle have the following angles?

55°, 55° and 80°

Answer

A triangle is possible only if the sum of its three angles is 180°.

Solving,

⇒ 55° + 55° + 80° = 190° ≠ 180°.

Since the sum is not 180°, such a triangle is not possible.

Hence, a triangle with angles 55°, 55° and 80° cannot be formed.

Can a triangle have the following angles?

33°, 74° and 73°

Answer

A triangle is possible only if the sum of its three angles is 180°.

Solving,

⇒ 33° + 74° + 73° = 180°.

Since the sum is 180°, such a triangle is possible.

Hence, a triangle with angles 33°, 74° and 73° can be formed.

Can a triangle have the following angles?

85°, 95° and 22°

Answer

A triangle is possible only if the sum of its three angles is 180°.

Solving,

⇒ 85° + 95° + 22° = 202° ≠ 180°.

Since the sum is not 180°, such a triangle is not possible.

Hence, a triangle with angles 85°, 95° and 22° cannot be formed.

Find x, if the angles of a triangle are :

x°, x°, x°

Answer

By the angle sum property of a triangle,

⇒ x° + x° + x° = 180°

⇒ 3x° = 180°

⇒ x° = $\dfrac{180\degree}{3}$

⇒ x° = 60°

⇒ x = 60

Hence, x = 60.

Find x, if the angles of a triangle are :

x°, 2x°, 2x°

Answer

By the angle sum property of a triangle,

⇒ x° + 2x° + 2x° = 180°

⇒ 5x° = 180°

⇒ x° = $\dfrac{180\degree}{5}$

⇒ x° = 36°

⇒ x = 36

Hence, x = 36.

Find x, if the angles of a triangle are :

2x°, 4x°, 6x°

Answer

By the angle sum property of a triangle,

⇒ 2x° + 4x° + 6x° = 180°

⇒ 12x° = 180°

⇒ x° = $\dfrac{180\degree}{12}$

⇒ x° = 15°

⇒ x = 15

Hence, x = 15.

One angle of a right-angled triangle is 70°. Find the other acute angle.

Answer

In a right-angled triangle, one angle is 90°. Let the other acute angle be x.

By the angle sum property of a triangle,

⇒ 90° + 70° + x = 180°

⇒ 160° + x = 180°

⇒ x = 180° − 160°

⇒ x = 20°

Hence, the other acute angle is 20°.

In △ABC, ∠A = ∠B = 62°; find ∠C.

Answer

By the angle sum property of a triangle,

⇒ ∠A + ∠B + ∠C = 180°

⇒ 62° + 62° + ∠C = 180°

⇒ 124° + ∠C = 180°

⇒ ∠C = 180° − 124°

⇒ ∠C = 56°

Hence, ∠C = 56°.

In △ABC, ∠B = ∠C and ∠A = 100°; find ∠B.

Answer

Given,

∠A = 100° and ∠B = ∠C. Let ∠B = ∠C = x.

By the angle sum property of a triangle,

⇒ ∠A + ∠B + ∠C = 180°

⇒ 100° + x + x = 180°

⇒ 100° + 2x = 180°

⇒ 2x = 180° − 100°

⇒ 2x = 80°

⇒ x = $\dfrac{80\degree}{2}$

⇒ x = 40°

⇒ ∠B = x = 40°

Hence, ∠B = 40°.

Find, giving reasons, the unknown marked angles in the triangle drawn below :

Answer

From the figure, in triangle ABC, ∠B = 30° and the side BC is produced to D, so ∠ACD = 110° is the exterior angle at C.

By the exterior angle property, an exterior angle of a triangle is equal to the sum of its two interior opposite angles.

⇒ ∠ACD = ∠A + ∠B

⇒ 110° = x + 30°

⇒ x = 110° − 30°

⇒ x = 80°

Hence, x = 80°.

Find, giving reasons, the unknown marked angles in the triangle drawn below :

Answer

From the figure, SQ is a straight line and ∠PQS = 115° is the exterior angle at Q, while the base angles ∠PQR and ∠PRQ are each equal to x.

Since ∠PQS and ∠PQR form a linear pair,

⇒ ∠PQS + ∠PQR = 180°

⇒ 115° + x = 180°

⇒ x = 180° − 115°

⇒ x = 65°

So the base angles ∠PQR = ∠PRQ = x = 65°.

By the angle sum property of a triangle,

In triangle PQR,

⇒ y + x + x = 180°

⇒ y + 65° + 65° = 180°

⇒ y + 130° = 180°

⇒ y = 180° − 130°

⇒ y = 50°

Hence, x = 65° and y = 50°.

Find, giving reasons, the unknown marked angles in the given figure :

Answer

From the figure,

∠XYM is an exterior angle of triangle and the two opposite interior angles are ∠YMZ and ∠YZM

Thus, 110° is the exterior angle of the triangle and the two interior opposite angles are 2x and 3x.

By the exterior angle property, an exterior angle of a triangle is equal to the sum of its two interior opposite angles.

⇒ 110° = 2x + 3x

⇒ 110° = 5x

⇒ x = $\dfrac{110\degree}{5}$

⇒ x = 22°

So, 2x = 44° and 3x = 66°.

Hence, the marked angles are 2x = 44° and 3x = 66°.

Classify the following triangle according to the measures of its angles :

Answer

From the figure, the angles of the triangle are 28°, 120° and 32°.

Since one of the angles, 120°, is more than 90°, it is an obtuse angle.

Hence, the given triangle is an obtuse-angled triangle.

Classify the following triangle according to the measures of its angles :

Answer

From the figure, the angles of the triangle are 80°, 70° and 30°.

Since each of the three angles is less than 90°, all the angles are acute.

Hence, the given triangle is an acute-angled triangle.

Classify the following triangle according to the measures of its angles :

Answer

From the figure, the angles of the triangle are 30°, 90° and 60°.

Since one of the angles is exactly 90°, it is a right angle.

Hence, the given triangle is a right-angled triangle.

Classify the following triangle according to the lengths of its sides :

Answer

From the figure, the sides of the triangle are 2.5 cm, 3 cm and 3 cm.

Two sides AC and BC are equal.

Hence, the given triangle is an isosceles triangle.

Classify the following triangle according to the lengths of its sides :

Answer

From the figure, the sides of the triangle are 3 cm, 4 cm and 2.5 cm.

Since all three sides are of different lengths, the given triangle is a scalene triangle.

Hence, the given triangle is a scalene triangle.

Classify the following triangle according to the lengths of its sides :

Answer

From the figure, the sides of the triangle are 3.5 cm, 3 cm and 2 cm.

Since all three sides are of different lengths, the given triangle is a scalene triangle.

Hence, the given triangle is a scalene triangle.

Classify the following triangle according to the lengths of its sides :

Answer

From the figure, the sides of the triangle are 3 cm, 3 cm and 3 cm.

Since all three sides are equal, the given triangle is an equilateral triangle.

Hence, the given triangle is an equilateral triangle.