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 measures of angles of ΔPQR are 2x , 3x and 4x .

[Given.]

Step: 2

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

[Triangle angle sum theorem.]

Step: 3

2x + 3x + 4x = 180°

Step: 4

[Simplify.]

Step: 5

[Simplify.]

Step: 6

[Squaring on both sides.]

Step: 7

Therefore, x = 400°.

Correct Answer is : 400°

Step: 1

Since the measures of the angles of the triangle are in the ratio 2 : 3 : 5, the measures of the angles of it can be taken as 2x , 3x and 5x .

Step: 2

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

[Triangle angle sum theorem.]

Step: 3

2 x + 3 x + 5 x = 180 °

Step: 4

10x = 180°

[Simplify.]

Step: 5

[Simplify.]

Step: 6

2 x = 2 × 18° = 36°,

3x = 3 × 18× = 54° and 5 x = 5 × 18° = 90°

3

Step: 7

So, the measures of the angles of the triangle are 36°
, 54°
and 90° and hence the triangle is right triangle.

Correct Answer is : a right triangle

Step: 1

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

Step: 2

90° + 44° + x = 180°

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

Step: 3

134°
+ x = 180°

[Add.]

Step: 4

[Simplify.]

Step: 5

Therefore, x = 46°.

Correct Answer is : 46°

Step: 1

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

Step: 2

So, ∠BAC + ∠ABC + ∠BCA = 180°.

Step: 3

[Substitute the given values of ∠BAC and ∠ABC.]

Step: 4

Step: 5

[Simplify.]

Step: 6

Therefore, the measure of ∠ BCA is 40°.

Correct Answer is : 40°

Step: 1

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

Step: 2

∠BCA + 64° + 76° = 180°

Step: 3

∠BCA + 140° = 180°

Step: 4

∠BCA = 180° - 140°

[Simplify.]

Step: 5

Therefore, the measure of ∠BCA is 40°.

Correct Answer is : 40°

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- Triangle