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The angles \[\angle {\text{ABC}}\] and $\angle {\text{ABD}}$ form a pair of linear angles. If $\angle {\text{ABC}}$= $2x + {10^ \circ }$ and $\angle {\text{ABD}}$=$8x - {30^ \circ }$. Find $\angle {\text{ABC}}$ and$\angle {\text{ABD}}$.

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Last updated date: 22nd Mar 2024
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MVSAT 2024
Answer
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Hint:
We are going to approach this problem by using the definition of a linear pair of angles. Linear pairs of angles are formed when two lines intersect each other at a single point. We are going to use one of the properties of linear pair of angles that “the sum of angles of a linear pair always equal to ${180^ \circ }$”
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Complete step-by-step answer:
It is given that the angles \[\angle {\text{ABC}}\] and $\angle {\text{ABD}}$ form a pair of linear angles.
Also it is given that $\angle {\text{ABC}}$= $2x + {10^ \circ }$ and $\angle {\text{ABD}}$=$8x - {30^ \circ }$.
Now we are going to find the value of the given angles \[\angle {\text{ABC}}\] and $\angle {\text{ABD}}$
Now we are going to use the property that the sum of angles of a linear pair always equal to ${180^ \circ }$
Therefore, $\angle ABC + \angle ABD = {180^ \circ }$.
We are going to substitute the given values on the above equation. Then we get,
$2x + {10^ \circ } + 8x - {30^ \circ } = {180^ \circ }$
We can only add or subtract the like terms. \[2x\]and \[8x\]are like terms, \[{10^ \circ }\] and ${30^ \circ }$ are like terms.
$(2x + 8x) + ({10^ \circ } - {30^ \circ }) = {180^ \circ }$
Now we are going to add and subtract the like terms. Then we get,
$10x - {20^ \circ } = {180^ \circ }$
Now we are going to keep like term on the same side. Then we get,
$10x = {180^ \circ } + {20^ \circ }$
$ \Rightarrow 10x = {200^ \circ }$
Now we are going to divide 10 on both sides. Then we get,
$x = {\dfrac{{200}}{{10}}^ \circ }$
Now we will divide 200 by 10. Then we get,
$x = {20^ \circ }$
Now we have found the value of$x = {20^ \circ }$.
We will substitute the value of x in the given value of $\angle ABC$and $\angle ABD$
$\angle ABC$$ = 2x + {10^ \circ }$
Now we substitute the value of $x$
$\angle ABC = $$2 \times {20^ \circ } + {10^ \circ } = {40^ \circ } + {10^ \circ } = {50^ \circ }$
$ \Rightarrow \angle ABC = $ ${50^ \circ }$
$\angle ABD = $$8x - {30^ \circ }$
Now we substitute the value of $x$
$\angle ABD = $$8 \times {20^ \circ } - {30^ \circ } = {160^ \circ } - {30^ \circ } = {130^ \circ }$
$ \Rightarrow \angle ABD$$ = {130^ \circ }$
Hence we found the given linear pair of angles \[\angle {\text{ABC = 5}}{{\text{0}}^ \circ }\] and $\angle {\text{ABD = 13}}{{\text{0}}^ \circ }$

Note:
We know that the sum of angles of a linear pair is always equal to ${180^ \circ }$.
$\angle ABC + \angle ABD = {50^ \circ } + {130^ \circ } = {180^ \circ }$
In this way, we can check whether the solution is correct or wrong.