Triangles

Let the third angle be x.

By the angle sum property of a triangle, the sum of all the angles of a triangle is 180°.

⇒ 72° + 63° + x = 180°

⇒ 135° + x = 180°

⇒ x = 180° − 135° = 45°

Hence, the third angle is 45°.

Given,

The angles of a triangle are in the ratio 2 : 3 : 5.

Let the angles of the triangle be 2x, 3x and 5x.

By the angle sum property of a triangle, the sum of all the angles of a triangle is 180°.

⇒ 2x + 3x + 5x = 180°

⇒ 10x = 180°

⇒ x = $\frac{180^{\circ}}{10}$

⇒ x = 18°

So, the angles are 2x = 2 × 18° = 36°, 3x = 3 × 18° = 54° and 5x = 5 × 18° = 90°.

Hence, the angles of the triangle are 36°, 54° and 90°.

Given,

The acute angles of a right triangle are in the ratio 1 : 2.

Let the two acute angles be x and 2x.

One angle of a right triangle measures 90°.

By the angle sum property of a triangle, the sum of all the angles of a triangle is 180°.

⇒ x + 2x + 90° = 180°

⇒ 3x + 90° = 180°

⇒ 3x = 180° − 90°

⇒ 3x = 90°

⇒ x = $\frac{90^{\circ}}{3}$

⇒ x = 30°

So, the two acute angles are x = 30° and 2x = 2 × 30° = 60°.

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

Let the other acute angle be x.

One angle of a right triangle measures 90° and one of its acute angles is 36°.

By the angle sum property of a triangle, the sum of all the angles of a triangle is 180°.

⇒ 90° + 36° + x = 180°

⇒ 126° + x = 180°

⇒ x = 180° − 126°

⇒ x = 54°

Hence, the other acute angle is 54°.

Let each of the two equal angles be x.

By the angle sum property of a triangle,

⇒ 106° + x + x = 180°

⇒ 2x = 180° − 106° = 74°

⇒ x = $\frac{74^{\circ}}{2}$

⇒ x = 37°

Hence, each of the equal angles is 37°.

Given,

Each of the two equal angles of an isosceles triangle is half the third angle.

Let the angles of the triangle be x, x and 2x.

By the angle sum property of a triangle,

⇒ x + x + 2x = 180°

⇒ 4x = 180°

⇒ x = $\frac{180^{\circ}}{4}$

⇒ x = 45°

So, the angles are x = 45°, x = 45° and 2x = 2 × 45° = 90°.

Hence, the angles of the triangle are 45°, 45° and 90°.

The given figure is a six-pointed star, which is made up of two triangles, △ACE and △BDF.

∠A, ∠C and ∠E are the three angles of △ACE, so by the angle sum property,

⇒ ∠A + ∠C + ∠E = 180° ... (1)

∠B, ∠D and ∠F are the three angles of △BDF, so by the angle sum property,

⇒ ∠B + ∠D + ∠F = 180° ...(2)

Adding (1) and (2),

⇒ (∠A + ∠C + ∠E) + (∠B + ∠D + ∠F) = 180° + 180°

⇒ ∠A + ∠B + ∠C + ∠D + ∠E + ∠F = 360°

Hence, ∠A + ∠B + ∠C + ∠D + ∠E + ∠F = 360°.

A triangle is acute-angled if each angle is less than 90°, right-angled if one angle is 90°, and obtuse-angled if one angle is more than 90°.

(i) One angle is 120°, which is more than 90°, so it is an obtuse-angled triangle.

(ii) One angle is 95°, which is more than 90°, so it is an obtuse-angled triangle.

(iii) One angle is 90°, so it is a right-angled triangle.

(iv) The angles are 85°, 60° and 35°, each less than 90°, so it is an acute-angled triangle.

(i) No. If each angle were less than 60°, the sum of the three angles would be less than 180°, which is not possible.

(ii) No. If each angle were greater than 60°, the sum of the three angles would be greater than 180°, which is not possible.

(iii) No. Two obtuse angles would each be more than 90°, so their sum alone would be more than 180°, which is not possible.

(iv) No. Two right angles would add up to 180°, leaving nothing for the third angle, which is not possible.

(v) Yes. Every triangle has at least two acute angles.

Three angles can be the angles of a triangle only if their sum is 180°.

(i) 50° + 30° + 100° = 180°. Yes.

(ii) 90° + 50° + 50° = 190° ≠ 180°. No.

(iii) 20° + 70° + 90° = 180°. Yes.

(iv) 45° + 90° + 45° = 180°. Yes.

(v) 65° + 65° + 65° = 195° ≠ 180°. No.

(vi) 120° + 50° + 10° = 180°. Yes.

A triangle is equilateral if all three sides are equal, isosceles if two sides are equal, and scalene if all sides are of different lengths.

(i) Two sides are equal (5 cm and 5 cm), so it is an isosceles triangle.

(ii) Two sides are equal (4 cm and 4 cm), so it is an isosceles triangle.

(iii) All three sides are equal (6 cm each), so it is an equilateral triangle.

(iv) All three sides are of different lengths (4.5 cm, 5 cm and 6 cm), so it is a scalene triangle.

Three line segments can form a triangle only if the sum of any two sides is greater than the third side.

(i) 5 + 4 = 9, which is not greater than 9. No.

(ii) 8 + 6 = 14 < 15. No.

(iii) 5 + 5 = 10 > 5 (and this holds for every pair). Yes.

(iv) 3.5 + 4.5 = 8 > 6.5, and every other pair also has a sum greater than the third side. Yes.

(v) 6 + 8 = 14 > 10, and every other pair also has a sum greater than the third side. Yes.

(vi) 3 + 3 = 6 < 7. No.

(i) No. Each angle of an equilateral triangle is 60°, so it can never have a 90° angle.

(ii) Yes. A triangle with angles 45°, 45° and 90° is both isosceles and right-angled.

(iii) Yes. A right triangle can have all three sides of different lengths, for example 3 cm, 4 cm and 5 cm.

(iv) No. A right triangle already has one 90° angle; an obtuse angle would make the sum of the angles exceed 180°.

(i) A triangle has 3 vertices, 3 sides and 3 angles.

(ii) The sum of the angles of a triangle is 180°.

(iii) Each angle of an equilateral triangle measures 60°.

(iv) The sum of any two sides of a triangle is greater than the third side.

(v) A right triangle cannot have an obtuse angle.

The sum of any two sides must be greater than the third side.

⇒ For (1): 5 + 4 = 9, not greater than 9.

⇒ For (2): 6 + 3 = 9 < 10.

⇒ For (3): 6 + 7 = 13 > 8, and every other pair also has a sum greater than the third side.

⇒ For (4): 5 + 5 = 10 < 11.

Only the lengths in (3) satisfy the condition.

Hence, option 3 is the correct option.

The sum of the angles of a triangle must be 180°.

⇒ In option (1): 30° + 40° + 50° = 120°.

⇒ In option (2): 40° + 50° + 60° = 150°.

⇒ In option (3): 50° + 60° + 70° = 180°.

⇒ In option (4): 55° + 65° + 75° = 195°.

Only the angles in option (3) add up to 180°.

Hence, option 3 is the correct option.

⇒ A triangle has 3 sides and 3 angles, so the total number of parts = 3 + 3 = 6.

Hence, option 4 is the correct option.

Given,

The angles of a triangle are in the ratio 3 : 4 : 5.

Let the angles of the triangle be 3x, 4x and 5x.

By the angle sum property of a triangle, the sum of all the angles of a triangle is 180°.

⇒ 3x + 4x + 5x = 180°

⇒ 12x = 180°

⇒ x = $\frac{180^{\circ}}{12}$

⇒ x = 15°

So, the smallest angle = 3x = 3 × 15° = 45°.

Hence, option 2 is the correct option.

Let the third angle be x.

It is given that the sum of the other two angles of the triangle is 60°.

By the angle sum property of a triangle, the sum of all the angles of a triangle is 180°.

⇒ 60° + x = 180°

⇒ x = 180° − 60°

⇒ x = 120°

So, the third angle is 120°.

Hence, option 4 is the correct option.

An equilateral triangle has all its three sides equal. Since the angles opposite to equal sides are equal, all its three angles are also equal.

Let each angle of the equilateral triangle be x.

By the angle sum property of a triangle, the sum of all the angles of a triangle is 180°.

⇒ x + x + x = 180°

⇒ 3x = 180°

⇒ x = $\frac{180^{\circ}}{3}$

⇒ x = 60°

So, each angle of an equilateral triangle measures 60°.

Hence, option 4 is the correct option.

In an isosceles triangle, the two base angles are equal. Since one base angle is 50°, the other base angle is also 50°.

Let the vertical angle be x.

By the angle sum property of a triangle, the sum of all the angles of a triangle is 180°.

⇒ 50° + 50° + x = 180°

⇒ 100° + x = 180°

⇒ x = 180° − 100°

⇒ x = 80°

So, the vertical angle is 80°.

Hence, option 3 is the correct option.

Let the third angle be x.

The other two angles of the triangle are 25° and 30°.

By the angle sum property of a triangle, the sum of all the angles of a triangle is 180°.

⇒ 25° + 30° + x = 180°

⇒ 55° + x = 180°

⇒ x = 180° − 55°

⇒ x = 125°

So, the third angle is 125°.

Hence, option 4 is the correct option.

⇒ A triangle whose sides are all of different lengths is a scalene triangle.

Hence, option 2 is the correct option.

⇒ The point P lies within the region enclosed by the triangle, so it is in the interior of △ABC.

Hence, option 3 is the correct option.

(i) The sum of any two sides of a triangle is greater than the third side.

(ii) The sum of the angles of a triangle is 180°.

(iii) A triangle has 3 vertices and 3 sides.

(iv) All angles of a scalene triangle are unequal.

(v) The medians of a triangle are concurrent.

(i) F — two obtuse angles would each be more than 90°, so their sum alone would exceed 180°.

(ii) T — every triangle has at least two acute angles.

(iii) F — two right angles would add up to 180°, leaving nothing for the third angle.

(iv) F — this is not always true; for example, in a triangle with angles 30°, 30° and 120°, the sum 30° + 30° = 60° is less than 120°.

(v) T — the third angle is 90°, so the two acute angles add up to 90°.

1. The length of the third side must be less than the sum and greater than the difference of the other two sides.

Sum of the two given sides = 18 cm + 14 cm = 32 cm

Difference of the two given sides = 18 cm − 14 cm = 4 cm

So, the length of the third side must be greater than 4 cm and less than 32 cm. Among the given options, only 5 cm satisfies this condition.

Hence, option (b) is the correct option.

2. The construction is possible only if the sum of any two sides is greater than the third side.

⇒ For (a): 3 + 5 = 8, which is not greater than 8.

⇒ For (b): 4 + 5 = 9 < 10.

⇒ For (c): 5 + 6 = 11 > 7, and every other pair also has a sum greater than the third side.

Only the sides given in option (c) satisfy the condition.

Hence, option (c) is the correct option.

3. The construction is possible only if the sum of the three angles is 180°.

⇒ For (a): 50° + 60° + 80° = 190°.

⇒ For (b): 40° + 50° + 60° = 150°.

⇒ For (c): 55° + 55° + 55° = 165°.

⇒ For (d): 35° + 45° + 100° = 180°.

Only the angles given in option (d) add up to 180°.

Hence, option (d) is the correct option.

4. A triangle can be constructed uniquely when two sides and the angle included between them are given.

⇒ In option (b), the two sides AB and BC are given along with ∠B, which is the angle included between them. This is the case of two sides and the included angle, so the construction is possible.

⇒ In option (a), ∠C is not the angle included between AB and BC, and in option (c), ∠A is not the angle included between AB and BC, so these do not give a determinate triangle.

Hence, option (b) is the correct option.

Since AD ⊥ BC, both △ADB and △ADC have a right angle at D. Also, △BAC has a right angle at A. So there are three right triangles in the figure: △ADB, △ADC and △BAC. So, Assertion (A) is true.

Reason (R) states that a triangle with one 90° angle is a right angled triangle, which is true.

Each of △ADB, △ADC and △BAC is counted as a right triangle exactly because it has one angle of 90°. So, Reason (R) is the correct explanation of Assertion (A).

Hence, option 1 is the correct option.

The point of intersection of the medians of a triangle is called the centroid, not the orthocentre. (The orthocentre is the point of intersection of the altitudes.) So, Assertion (A) is false.

A triangle has three medians, one from each vertex to the mid-point of the opposite side. So, Reason (R) is true.

Hence, option 4 is the correct option.