Step: 1

Let 4x , 5x and 6x be the angles.

Step: 2

4x + 5x + 6x = 180^{o}

[Triangle angle sum theorem.]

Step: 3

15x = 180^{o}

x = 1 8 0 ° 1 5

x = 12^{o}

[Divide each side by 15.]

Step: 4

So, 4x = 4 × 12^{o} = 48^{o}

5x = 5 × 12^{o} = 60^{o}

6x = 6 × 12^{o} = 72^{o}

5

6

[Substitute x = 12^{o}.]

Step: 5

The angles of the triangle are 48^{o}, 60^{o} and 72^{o} .

Correct Answer is : 48^{o}, 60^{o}, 72^{o}

(i) One arm of

(ii).

(iii). m

(iv) m

(v).

Step: 1

The two angles can be with two different arms. Statement (i) need not be correct.

Step: 2

The two angles can be the angles of the same triangle since sum of the angles of a triangle is 180 degrees.

Step: 3

(ii), (iii) and (iv) are correct.

Correct Answer is : (ii), (iii) and (iv)

Step: 1

[Linear pair.]

Step: 2

[Simplify.]

Step: 3

The sum of the measures of the angles of a triangle is 180.

[Triangle angle sum theorem.]

Step: 4

60 + y + z = 180

[From step 3.]

Step: 5

60 + 60 + z = 180

[Substitute 60 for y .]

Step: 6

[Simplify.]

Step: 7

So, 2y + 3z = 120 + 180 = 300

[Substitute 60 for y and z .]

Correct Answer is : 300

Step: 1

The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles.

[Exterior angle theorem.]

Step: 2

[From step 1.]

Step: 3

The sum of the measure of the angles of a triangle is 180.

[Triangle angle-sum theorem.]

Step: 4

[From step 3.]

Step: 5

[Substitute y = 2x .]

Step: 6

5x = 180 ⇒ x = 36

[Simplify.]

Step: 7

[From step 2.]

Step: 8

Hence the value of x is 36 and y is 72.

Correct Answer is : 36 and 72

Step: 1

Since the measures of the angles of the triangle are in the ratio 2: 4: 6, the measures of the angles of triangle can be taken as 2x , 4x and 6x .

Step: 2

The sum of the measure of angles of a triangle is 180.

[Triangle angle sum theorem.]

Step: 3

2x + 4x + 6x =180

[ From steps 1 and 2 ]

Step: 4

12x = 180

[Simplify]

Step: 5

[solve for x ]

Step: 6

The measures of the angles of triangle are,

Step: 7

2x = 2(15) = 30,

Step: 8

4x =4(15) = 60 and

Step: 9

6x = 6(15) = 90

Step: 10

Therefore, the largest angle of a triangle = 90°.

Correct Answer is : 90°

Step: 1

The sum of all the three angles in a triangle is 180°.

[Angle sum theorem of a triangle.]

Step: 2

The measures of angles of a triangle are perfect squares.

[Given.]

Step: 3

If the angles of a triangle are a °, b °, and c °, then a + b + c = 180°, where a , b , and c are perfect squares.

Step: 4

Let the angles be a = 16°, b = 64°, and c = 100°.

Step: 5

⇒ 16 + 64 + 100 = 180

[16 = 4^{2}, 64 = 8^{2}, and 100 = 10^{2}.]

Step: 6

Here, the largest angle is 100 and the smallest angle is 16.

Step: 7

So, the difference between the largest and the smallest angles = 100 - 16 = 84°.

Correct Answer is : 84°

Step: 1

The sum of all the angle measures of a triangle is 180°.

Step: 2

90° + 62° + y ° = 180°

[Equate the sum of angles of the triangle to 180°.]

Step: 3

152° + y ° = 180°

[Add.]

Step: 4

[Subtract 152° from each side.]

Correct Answer is : 28°

Step: 1

Let 5x , 6x and 7x be the angles.

Step: 2

Sum of the measures of the angles in a triangle = 180°

Step: 3

5x + 6x + 7x = 180°

18x = 180°

x = 1 8 0 ° 1 8

x = 10°

18

[Divide each side by 18.]

Step: 4

5x = 5 × 10° = 50°

6x = 6 × 10° = 60°

7x = 7 × 10° = 70°

6

7

[Substitute x = 10° .]

Step: 5

The measures of the angles are 50°, 60° and 70° .

Correct Answer is : 50°, 60°, 70°

Step: 1

The centroid of a triangle divides the medians in the ratio 2:1. That is A G : G D = 2 : 1

Step: 2

Step: 3

Therefore, the length of the median from the centroid to the opposite side of the vertex is equal to half the length of the median from the vertex to the centroid.

Correct Answer is : The length of the median from the centroid to the opposite side of the vertex is equal to the length of the median from the vertex to the centroid.

Step: 1

Step: 2

The centroid of a triangle divides the medians in the ratio 2:1.

Step: 3

That is AG : GD = 2 : 1.

Step: 4

Step: 5

AD = AG + GD = 2 GD + GD = 3 GD.

Step: 6

⇒ A D G D = 3 1

Step: 7

Therefore,AD : GD = 3 : 1.

Correct Answer is : 3 : 1

Step: 1

In an equilateral triangle, the median from any vertex to the opposite side will be perpendicular to that side.

Step: 2

In an isosceles triangle, the median from the vertex containing the congruent sides is perpendicular to the base.

Step: 3

So, in both isosceles triangle and equilateral triangle, at least one median coincides with an altitude.

Correct Answer is : At least one median coincides with an altitude in both isosceles and equilateral triangles.

Step: 1

PQ || AB, QR || BC, PR ||AC

[Mid-segment theorem.]

Step: 2

PQ = 1 2 AB ⇒ AB = 2 PQ = 2 × 8 = 16 cm.

Step: 3

QR = 1 2 BC ⇒ BC = 2 QR = 2 × 10 = 20 cm.

Step: 4

PR = 1 2 AC ⇒ AC = 2 PR = 2 × 12 = 24 cm.

Correct Answer is : 16 cm, 20 cm and 24 cm respectively

Step: 1

ΔPQR ~ ΔCQA

[A and C are midpoints of QR and PQ and AC is the midsegment.]

Step: 2

⇒ Q D Q B = 1 2 ⇒ Q D Q D + D B = 1 2

[Ratio of the sides of the similar triangles ΔCQA and ΔPQR is 1 : 2.]

Step: 3

2QD = QD + DB ⇒ QD = DB = 15 cm

[DB = BD = 15 cm.]

Step: 4

QB = QD + DB = 15 cm + 15 cm = 30 cm

Step: 5

QG = 2 3 × QB = 2 3 × 30 = 20 cm

[G is the point of intersection of the medians, which divides the median in 2 : 1 ratio.]

Step: 6

GD = QG - QD = 20 - 15 = 5 cm

Correct Answer is : 5 cm

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