Question

ratio 2:5. Find the angles of the triangle.
One of the exterior angle of a triangle is 105° and the interior opposite angles are in ratio 2:5 find the angle of the triangle​

Answer 1

✬ Angles = 30° ,75° & 75° ✬

Step-by-step explanation:

Given:

• Measure of exterior angle of triangle is 105°.
• Ratio of opposite angles of triangle is 2 : 5.

To Find:

• Measure of angles of triangle ?

Solution: Let x be the common in given ratios and ABC be a triangle.

Here in triangle ABC we have

• ∠ACD = exterior angle of ∆.

• ∠BAC : ∠CBA = 2 : 5.

[ Remind a theorem ]

• The sum of exterior angle of a triangle is equal to the sum of its opposite interior angles.

• ∠BAC & ∠CBA are opposite interior angles.

$\implies{\rm }$ ∠BAC + ∠CBA = ∠ACD

$\implies{\rm }$ 2x + 5x = 105°

$\implies{\rm }$ 7x = 105°

$\implies{\rm }$ x = 105°/7

$\implies{\rm }$ x = 15°

So , measure of

• ∠BAC = 2 × 15 = 30°
• ∠CBA = 5 × 15 = 75°

Now by using a property of triangle.

[ Angle sum property ]

• It states that sum of all interior angles of a ∆ is equal to 180°.

➟ ∠BAC + ∠CBA + ∠ACB = 180°

➟ 30° + 75° + ∠ACB = 180°

➟ ∠ACB = 180° – 105°

➟ ∠ACB = 75°

Answer 2

We have the property that, in a triangle exterior angle is equal to the sum of interior opposite angles.

Given, the interior opposite angles to the exterior angle 105° are in the ration 2:5

...2x + 5x = 1050

7x=105° =

X = 15°

Therefore the interior opposite angles to the angle 105° are 2 x = 2(15) = 30° and 5x = 5(15) = 75° =

Let the third angle of the triangle be C We have the sum of interior angles of a triangle is

180°

.:. 30° + 75° + C = 1800

C = 180° - 75° 30° = 180° - 105°

C = 75°

hence the angles are 30⁰,75⁰,75⁰