[tex]\bold{\huge{\underline{ Solution }}}[/tex]
Given :-
- Here, we have given one quadrilateral that is quadrilateral ABCD
- We also have given the angles of quadrilateral that is ( 6x + 5)° , ( 9x - 10)° , 80° and a right angle
To Find :-
Here, we have to find the value of x
Let's Begin :-
We have given quadrilateral ABCD here , whose angles are as follows
- Angle A = ( 6x + 5)°
- Angle B = ( 9x - 10)°
- Angle C = 80°
- Angle D = A right angled triangle
[ The measure of right angled triangle is 90° ]
We know that,
- Sum of the angles of quadrilateral is equal to 360°
That is
[tex]\bold{\angle{ A + }}{\bold{\angle{B +}}}{\bold{\angle{C + }}}{\bold{\angle{D + = 360{\degree}}}}[/tex]
Subsitute the required values
[tex]\sf{ ( 6x + 5){\degree} + 80{\degree}+ 90{\degree} + (9x - 10){\degree} = 360{\degree}}[/tex]
[tex]\sf{ 6x + 5 + 170{\degree} + 9x - 10 = 360{\degree}}[/tex]
[tex]\sf{ 15x - 5 = 360{\degree} - 170{\degree}}[/tex]
[tex]\sf{ 15x - 5 = 190 {\degree}}[/tex]
[tex]\sf{ 15x = 190 {\degree} + 5 }[/tex]
[tex]\sf{ 15x = 195 {\degree}}[/tex]
[tex]\sf{ x = }{\sf{\dfrac{ 195}{15}}}[/tex]
[tex]\sf{ x = }{\sf{\cancel{\dfrac{ 195}{15}}}}[/tex]
[tex]\bold{ x = 13 }[/tex]
Hence, The value of x is 13 .
[tex]\bold{\huge{\underline{\red{ Verification }}}}[/tex]
Measure of Angle A
[tex]\sf{ = (6x + 5){\degree}}[/tex]
[tex]\sf{ = (6(13) + 5 ){\degree}}[/tex]
[tex]\sf{ = (78 + 5 ){\degree}}[/tex]
[tex]\sf{ = 83{\degree}}[/tex]
Measure of Angle D
[tex]\sf{ = (9x - 10){\degree}}[/tex]
[tex]\sf{ = (9(13) - 10){\degree}}[/tex]
[tex]\sf{ = (117 - 10 ){\degree}}[/tex]
[tex]\sf{ = 107{\degree}}[/tex]
Now, we know that,
- Sum of angles of triangles is equal to 360°
That is,
[tex]\sf{ 83{\degree} + 80{\degree}+ 90{\degree} + 107 {\degree} = 360{\degree}}[/tex]
[tex]\sf{ 163{\degree}+ 90{\degree} + 107 {\degree} = 360{\degree}}[/tex]
[tex]\sf{ 253{\degree}+ 107 {\degree} = 360{\degree}}[/tex]
[tex]\sf{ 253{\degree}+ 107 {\degree} = 360{\degree}}[/tex]
[tex]\bold{ 360{\degree} = 360{\degree}}[/tex]
[tex]\bold{ LHS = RHS }[/tex]
Hence, Proved.
