Triangles — Multiple Choice Questions

In a △ABC, AB = AC and ∠B = 50°. Then ∠A =

1. 50°

2. 80°

3. 100°

4. 105°

Answer

In △ABC,

AB = AC

⇒ ∠B = ∠C = 50° (Angles opposite to equal sides in a triangle are equal)

By angle sum property of triangle,

⇒ ∠A + ∠B + ∠C = 180°

⇒ ∠A + 50° + 50° = 180°

⇒ ∠A + 100° = 180°

⇒ ∠A = 180° - 100°

⇒ ∠A = 80°.

Hence, option 2 is the correct option.

In a △PQR, ∠Q = 50°, ∠R = 65° and QR = 4 cm. Then PQ =

1. 6 cm

2. 5 cm

3. 4 cm

4. 3 cm

Answer

In △PQR,

By angle sum property of triangle,

⇒ ∠P + ∠Q + ∠R = 180°

⇒ ∠P + 50° + 65° = 180°

⇒ ∠P + 115° = 180°

⇒ ∠P = 180° - 115°

⇒ ∠P = 65°

Since, ∠P = ∠R = 65°

⇒ PQ = QR = 4 cm (Sides opposite to equal angles in a triangle are equal)

Hence, option 3 is the correct option.

In the adjoining figure, ∠ABC = 90°, ∠BCA = 50° and BD ⊥ AC. Then ∠ABD =

1. 30°

2. 40°

3. 50°

4. 60°

Answer

In △ABC,

By angle sum property of triangle,

⇒ ∠A + ∠B + ∠C = 180°

⇒ ∠A + 90° + 50° = 180°

⇒ ∠A + 140° = 180°

⇒ ∠A = 180° - 140°

⇒ ∠A = 40°.

In △ABD,

By angle sum property of triangle,

⇒ ∠A + ∠ABD + ∠D = 180°

⇒ 40° + ∠ABD + 90° = 180°

⇒ ∠ABD + 130° = 180°

⇒ ∠ABD = 180° - 130°

⇒ ∠ABD = 50°.

Hence, option 3 is the correct option.

In the adjoining figure, AB = AC and BD = CD. Then, ∠ADC =

1. 60°

2. 75°

3. 90°

4. 100°

Answer

In △ABD and △ACD,

⇒ AD = AD (Common side)

⇒ BD = CD (Given)

⇒ AB = AC (Given)

∴ △ABD ≅ △ACD (By S.S.S axiom)

⇒ ∠ADB = ∠ADC = x (let) (Corresponding parts of congruent triangles are equal)

From figure,

⇒ ∠ADB + ∠ADC = 180° (Linear pair)

⇒ x + x = 180°

⇒ 2x = 180°

⇒ x = $\dfrac{180°}{2}$

⇒ x = 90°

⇒ ∠ADB = ∠ADC = 90°.

Hence, option 3 is the correct option.

In the adjoining figure, AB = AC, BD = CD, ∠BAD = 32°, ∠BDC = 56°, ∠CAD = 2x° and ∠BDA = (x + y)°. The values of x and y will be :

1. x = 10, y = 16

2. x = 16, y = 12

3. x = 18, y = 8

4. x = 12, y = 16

Answer

Join BC.

In △BDC,

BD = DC

∠DBC = ∠DCB = a (let)

∴ ∠DBC + ∠DCB + ∠BDC = 180°

⇒ a + a + 56° = 180°

⇒ 2a = 180° - 56°

⇒ 2a = 124°

⇒ a = $\dfrac{124°}{2}$

⇒ a = 62°

⇒ ∠DBC = ∠DCB = 62°

In △ABC,

AB = AC

∠ABC = ∠ACB

∠ABD = ∠ABC + ∠DBC ....(1)

∠ACD = ∠ACB + ∠DCB

⇒ ∠ACD = ∠ABC + ∠DBC ....(2)

From eq.(1) and (2), we have :

⇒ ∠ABD = ∠ACD

In △ABC,

∴ ∠A + ∠ABC + ∠ACB = 180°

⇒ ∠BAD + ∠CAD + ∠ABC + ∠ABC = 180°

⇒ 32° + 2x° + 2∠ABC = 180°

⇒ 2∠ABC = 180° - 32° - 2x°

⇒ 2∠ABC = 148° - 2x°

⇒ ∠ABC = 74° - x°

Substituting value of ∠ABC in eq.(1):

⇒ ∠ABD = ∠ABC + ∠DBC

⇒ ∠ABD = 74° - x° + 62°

⇒ ∠ABD = 136° - x°

In △ABD,

⇒ ∠ABD + ∠BAD + ∠BDA = 180°

⇒ 136° - x° + 32° + x° + y° = 180°

⇒ 168° + y° = 180°

⇒ y° = 180° - 168°

⇒ y° = 12°

⇒ y = 12.

In an isosceles triangle BDC,

BD = CD

We know that,

Perpendicular drawn from the vertex of an isosceles triangle bisects the base.

⇒ ∠DOB = ∠DOC = 90°

In △DOB,

⇒ ∠DBO + ∠DOB + ∠BDO = 180°

⇒ 62° + 90° + x° + y° = 180°

⇒ 152° + x° + 12° = 180°

⇒ 164° + x° = 180°

⇒ x° = 180° - 164°

⇒ x° = 16°

⇒ x = 16.

Hence, option 2 is the correct option.

In the figure, AB = AC and DB = DC. ∠ABD : ∠ACD is :

1. 1 : 2

2. 2 : 1

3. 1 : 1

4. 1 : 3

Answer

In △ABC,

AB = AC

⇒ ∠B = ∠C = x ....(1) (let) (Angles opposite to equal sides in a triangle are equal)

In △BDC,

BD = CD

⇒ ∠DBC = ∠DCB = y ....(2) (let) (Angles opposite to equal sides in a triangle are equal)

Subtracting eq.(2) from (1), we have :

⇒ ∠B - ∠DBC = ∠C - ∠DCB

⇒ ∠ABD = ∠ACD

⇒ ∠ABD : ∠ACD = 1 : 1.

Hence, option 3 is the correct option.

In right triangles ABC and DEF, if hypotenuse AB = EF and side AC = DE, then △ABC is congruent to:

1. △FED

2. △EFD

3. △DEF

4. △FDE

Answer

In △ABC and △EFD,

⇒ AB = EF (Given)

⇒ AC = DE (Given)

⇒ ∠C = ∠D (Both equal to 90°)

∴ △ABC ≅ △EFD (By R.H.S axiom)

Hence, option 2 is the correct option.

ABC is a right angled triangle whose hypotenuse is AC. If AB : BC = 3 : 4, then the smallest angle of the triangle is :

1. ∠A

2. ∠B

3. ∠C

4. None of these

Answer

In △ABC,

Given,

AB : BC = 3 : 4

Let, AB = 3x and BC = 4x

By pythagoras theorem,

⇒ AC² = AB² + BC²

⇒ AC² = (3x)² + (4x)²

⇒ AC² = 9x² + 16x²

⇒ AC² = 25x²

⇒ AC = $\sqrt{25\text{x}^2}$

⇒ AC = 5x.

We know that,

The shortest angle of a triangle has the shortest side opposite to it.

Since, side AB is shortest, thus ∠C is the shortest angle.

Hence, option 3 is the correct option.

If a, b, c be the lengths of the sides of a triangle, then :

1. a = b + c

2. a < b + c

3. a > b + c

4. a < b - c

Answer

We know that,

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

⇒ a < b + c

Hence, option 2 is the correct option.

In a △ABC, AB > BC > CA. Then :

1. AB - BC < CA

2. AB - BC > CA

3. AB + BC < CA

4. None of these

Answer

We know that,

Difference between any two sides of a triangle must be less than the third side.

In △ABC,

AB - BC < CA

Hence, option 1 is the correct option.

In a △ABC, ∠A = 40° and ∠B = 60°. The longest side of the triangle is:

1. AB

2. BC

3. CA

4. None of these

Answer

In △ABC,

By angle sum property of triangle,

⇒ ∠A + ∠B + ∠C = 180°

⇒ 40° + 60° + ∠C = 180°

⇒ ∠C + 100° = 180°

⇒ ∠C = 180° - 100°

⇒ ∠C = 80°

We know that,

The longest side of a triangle has the largest angle opposite to it.

Since, angle C is the greatest, thus AB is longest side of the triangle.

Hence, option 1 is the correct option.

In a △ABC, AB = 6 cm, BC = 7 cm and CA = 8 cm. The smallest angle of the triangle is :

1. ∠A

2. ∠B

3. ∠C

4. None of these

Answer

We know that,

The smallest angle of a triangle has the smallest side opposite to it.

Since, AB is the smallest side thus ∠C is the smallest angle of the triangle.

Hence, option 3 is the correct option.

In a △ABC, 2∠A = 3∠B and ∠C = 100°. The correct ascending order of sides of the triangle is :

1. AC < BC < AB

2. BC < AC < AB

3. AB < AC < AB

4. BC < AB < AC

Answer

Given,

2∠A = 3∠B

⇒ ∠A : ∠B = 3 : 2

⇒ ∠A = 3x° and ∠B = 2x°

In △ABC,

By angle sum property of triangle,

⇒ ∠A + ∠B + ∠C = 180°

⇒ 3x° + 2x° + 100° = 180°

⇒ 5x° = 180° - 100°

⇒ 5x° = 80°

⇒ x° = $\dfrac{80°}{5}$

⇒ x° = 16°

⇒ ∠A = 3x° = 3 × 16° = 48°

⇒ ∠B = 2x° = 2 × 16° = 32°

We know that,

The longest side of a triangle has the largest angle opposite to it.

Since, C is the largest angle, thus AB is longest side of the triangle.

The shortest side of a triangle has the smallest angle opposite to it.

Since, B is the smallest angle, thus AC is smallest side of the triangle.

∴ AC < BC < AB.

Hence, option 1 is the correct option.

The angles of a triangle are 5(x - 4)°, (4x + 5)° and (x + 25)°, then the value of x is :

1. 17

2. 18

3. 19

4. 21

Answer

By angle sum property of triangle,

⇒ 5(x - 4)° + (4x + 5)° + (x + 25)° = 180°

⇒ 5x° - 20° + 4x° + 5° + x° + 25° = 180°

⇒ 10x° + 10° = 180°

⇒ 10x° = 180° - 10°

⇒ 10x° = 170°

⇒ x° = $\dfrac{170°}{10}$

⇒ x° = 17°

⇒ x = 17.

Hence, option 1 is the correct option.

In △ABC, AB > AC and D is any point on BC, then, AB is :

1. < DC

2. < AD

3. = BC

4. > AD

Answer

In △ABC,

We know that,

The larger angle of a triangle has the longer side opposite to it.

⇒ AB > AC

⇒ ∠ACB > ∠ABC ....(1)

From figure,

⇒ ∠ADB > ∠ACD (exterior angle of a triangle is greater than interior opposite angle)

⇒ ∠ADB > ∠ACB ....(2)

From eq.(1) and (2), we have:

⇒ ∠ADB > ∠ABC

⇒ ∠ADB > ∠ABD

⇒ AB > AD (larger angle of a triangle has the larger side opposite to it).

Hence, option 4 is the correct option.