Question

.. ABCD is a quadrilateral. AO and BO are the angle bisectors of angle A and B which meet at O. If angle C = 70°, angle D= 50°, find angle AOB.​

Answer 1

Answer.

∠C = 70° , ∠D = 50°

$\fbox{\underline\orange{To~Find~:}}$

We have to find ∠AOB .

$\fbox{\underline\green{Solution~:}}$

$\sf \: {Let \: ∠A \: and \: ∠B =} x$

We know that sum of all sides of Quadrilateral is 360° .

x + x + 70° + 50° = 360°

2x + 120° = 360°

2x = 360° - 120°

2x = 240°

$\sf \: x = \frac{240°}{2}$

x = 120°

In ∠AOB

∠A = 60° , ∠B = 60°

$\bold{Let , ∠O =}{ x}$

We know that sum of all sides in a Triangle is 180° .

$\sf ∠A + ∠B + ∠O = 180°\\ \\ 60° + 60° + \sf x = 180°\\ \\ 120° + \sf x = 180°\\ \\ \sf x = 180° - 120°\\ \\ \sf \implies\: \bold{ x = 60°}$

⠀⠀⠀⠀ ⠀⠀

$\boxed{\underline\red{ \sf \: x = 60°~Ans.}}$

Hence This is Answer.

Answer 2

Question :-

ABCD is a quadrilateral. AO and BO are the angle bisectors of angle A and B which meet at O. If angle C = 70°, angle D= 50°, find angle AOB.

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Solution :-

➲ Given Information :-

• AO ➢ Angle bisector of ∠A
• BO ➢ Angle bisector of ∠B
• AO & BO meet at point O
• Measure of ∠C ➢ 70°
• Measure of ∠D ➢ 50°

➲ To Find :-

• Measure of ∠AOB

➲ Concept :-

• Quadrilaterals

➲ Formula Used :-

1. $\boxed{ \boxed{ \bf{ \red{sum \: of \: all \: angles \: of \: a \: quadrilateral} = \blue{360 \degree}}}}$
2. $\boxed{ \boxed{ \bf{ \red{sum \: of \: all \: angles \: of \: a \: triangle} = \blue{180 \degree}}}}$

➲ Explanation :-

• First of all, we'll assume Measure of ∠A & ∠B as 'x'. Then we'll substitute all the values in the first formula provided above. After some minute calculations, we'll get the value of 'x'. The to find the measure of ∠AOB, well substitute the values and use the second formula mentioned above. After some minute calculations, we'll get the value of our ∠AOB. Now let's proceed towards our calculation.

➲ Calculation :-

Assuming the measure of ∠A & ∠B as 'x'

Using the first formula, We get,

$\boxed{ \boxed{ \bf{ \red{sum \: of \: all \: angles \: of \: a \: quadrilateral} = \blue{360 \degree}}}}$

Substituting the values given in the formula mentioned above, We get,

$\rm{:\implies \red{∠A + ∠B + ∠C + ∠D} = \blue{360 \degree}}$

Substituting the values, We get,

$\rm:\implies { \red{x + x + 70° + 50°= \blue{360 \degree}}}$

Adding up all the terms, We get,

$\rm:\implies { \red{2x + 120°= \blue{360 \degree}}}$

Transposing 120° to Right Hand Side of the equation, We get,

$\rm:\implies{ \red{ 2x = \blue{360 \degree - 120\degree}}}$

Subtracting the terms in Right Hand Side of the equation, We get,

$\rm:\implies{ \red{ 2x = \blue{240 \degree}}}$

Now, transposing '2' to the denominator in Right Hand Side of the equation, We get,

$\rm:\implies{ \red{x }= \blue{ \dfrac{240}{2} \degree}}$

Cancelling & Calculating further, We get,

$\bf:\implies\boxed{{ \boxed{\red{\rm x }= \blue{120 \degree}}}}$

Now, In ∠AOB

We now know, ∠A + ∠B = 120°

:⇒ x + x = 120°

:⇒ 2x = 120°

∴ ∠A = 60° , ∠B = 60°

Now, assuming ∠O as 'x'

Using the second formula, mentioned above, We get,

$\boxed{ \boxed{ \bf{ \red{sum \: of \: all \: angles \: of \: a \: triangle} = \blue{180 \degree}}}}$

Substituting the values given in the formula mentioned above, We get,

$\rm:\implies \red{∠A + ∠B + ∠O }= \blue{180°}$

Substituting the values, We get,

$\rm:\implies{\red {60° + 60° + x }=\blue{ 180°}}$

Adding up terms of Left Hand Side of the equation, We get,

$\rm:\implies{\red{120° + x }= \blue{180°}}$

Transposing 120° to Right Hand Side of the equation, We get,

$\rm:\implies{ \red{x} =\blue{ 180° - 120°}}$

Subtracting, We get,

$\rm:\implies{ \red x = \blue{60°}}$

$\therefore\boxed{\boxed{\red{ \bf \: ∠AOB }=\blue{ 60°}}}$

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Final Answer :-

• Measure of ∠AOB is 60°

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Note :-

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