Question

one of the exterior angles of the triangle is 105 and the interior opposite Angles are in the ratio 2:3. find the angles of the triangle?​

Answer 1

According to exterior angle theorem:

Exterior angle of a triangle is equal to the sum of the 2 interior opposite angles.

Let x be the common ratio. Then the angles are 2x° and 3x°.

2x + 3x = 105.

5x = 105.

x = 105/5.

x = 21°.

2x = 2(21) = 42°.

3x = 3(21) = 63°.

According to angle sum property:

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

Let the unknown angle be y.

y + 42 + 63 = 180.

y + 105 = 180.

y = 180 - 105.

y = 75°.

Therefore, the angles of the triangle are, 42°,63° and 75°.

Answer 2

$\mathfrak{\large{\underline{\underline{Answer :-}}}}$

The Measure of the angles in the triangles are - 42°, 63° and 75°.

$\mathfrak{\large{\underline{\underline{Explanation :-}}}}$

Given :

The Exterior Angle of the Triangle = 105°

Ratio of the angles = 2 : 3

To find :

The Measure of the angles in the the triangle.

Solution :-

Refer the attachment.

According to the property of triangle :

Sum of interior opposite Angles in the triangle = measure of the exterior Angle.

Consider one Angle as 2x and Second angle as 3x

$\tt{\implies} \: 2x + 3x = 105$

$\tt{\implies} \: 5x = 105$

$\tt{\implies} \: x = \dfrac{105}{5}$

$\tt{\implies} \: x = 21$

Value of 2x

$\tt{\implies} \:21 \times 2$

$\tt{\implies} \: 42^{\circ}$

Value of 3x

$\tt{\implies} \: 3 \times 21$

$\tt{\implies} \: 63^{\circ}$

Now, as we got the measures of two Angle, we can now find tgr Third angles by Angle sum property of triangle.

Angle Sum Property = Sum of all Angles in the triangle sum up and make 180°

Consider the Third Angle as z

$\tt{\implies} \: 42^{\circ} + 63^{\circ} + z = 180 ^{\circ}$

$\tt{\implies} \:105 + z = 180$

$\tt{\implies} \:z = 180 - 105$

$\tt{\implies} \:z = 75 ^{\circ}$

$\therefore$ The Measure of the angles in the triangles are - 42°, 63° and 75°.