Question

$\huge \mathfrak \red{Question}$​

Answer 1

Answer:

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 :-

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

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

➲ 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 :-

Please scroll from right to left to see the whole solution.

Answer 2

Step-by-step explanation:

Answer:

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.

▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂▂

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 :-

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

sumofallanglesofaquadrilateral=360°

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

sumofallanglesofatriangle=180°

➲ 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}}}}

sumofallanglesofaquadrilateral=360°

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

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

Substituting the values, We get,

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

Adding up all the terms, We get,

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

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

\rm:\implies{ \red{ 2x = \blue{360 \degree - 120\degree}}}:⟹2x=360°−120°

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

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

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

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

2

240

°

Cancelling & Calculating further, We get,

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

x=120°

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}}}}

sumofallanglesofatriangle=180°

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

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

Substituting the values, We get,

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

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

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

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

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

Subtracting, We get,

\rm:\implies{ \red x = \blue{60°}}:⟹x=60°

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

∠AOB=60°

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Final Answer :-

Measure of ∠AOB is 60°

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Note :-

Please scroll from right to left to see the whole solution.