Question

One of the exterior angles of a triangle is 70° and the interior opposite angles are in the ratio
3:4. Find the angles of the triangle.​

Answer 1

Answer:

Explanation:

let the opposite angles be 3x and 4x  

then

exterior angle = sum of opposite interior angles

implies  

70 = 3x + 4x

⇒7x = 70

⇒x=70÷7

⇒x=10

so opposite angles are

3x = 3×10 = 30

4x = 4×10 = 40

Answer 2

Given:

• One of the exterior angles of a triangle is 70° and the interior opposite angles are in the ratio 3:4.

To find:

• The angles of triangle.

Solution:

⠀⠀Exterior angle of the triangle = 70°

❍ Let the interior angles of the triangle be 3x and 4x

We know that,

❍ Exterior angle property of a triangle :

• If a side of a triangle is produced, then the exterior angleso formed is equal to sum of two interior opposite angles.

$\dashrightarrow\sf \: \: \: \: 3x + 4x + = 70^\circ \\\ \\\dashrightarrow\sf \: \: \: \: \: \: \: \: \: \: 7x = 70^\circ \\\\\dashrightarrow \: \: \: \: \: \: \: \sf x = \sf\cancel\dfrac{70}{7} \\\\\ \: \: \: \: \: \: \: \dashrightarrow \: \: \: \pmb x = \underline{\boxed{\pmb{\pink{10 {}^{ \circ} }}}}$

Hence,

• 3x = 6 × 10° = 30°
• 4x = 4 × 10° = 40°

⠀⠀⠀

$\: \:\implies\sf \Big(First\;angle\Big) +\Big(Second\;angle\Big)+\Big(Third\;angle\Big) = Sum \: of \: the \: angles \: of \: the \: triangle\\\\\\\ \: \: \: \: \: \: \implies\sf 30^\circ + 40^\circ + third \: angle = 180^\circ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\\\\\\implies\sf third \: angle = 180^\circ - 70 {}^{ \circ} = 110 {}^{ \circ} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

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❍ VERIFICATION:

$\: \: \: \: \: \: \: \: \: \:\implies\sf \Big(First\;angle\Big) +\Big(Second\;angle\Big)+\Big(Third\;angle\Big) = Sum \: of \: the \: angles \: of \: the \: triangle\\\\\\\implies\sf 30^\circ + 40^\circ + 110^\circ = 180^\circ \: \: \: \: \ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\\\\\\implies\sf 180^\circ = 180^\circ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \ {\underline{\boxed{\pmb{{L.H.S = R.H.S}}}}} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀HENCE VERIFIED!!