Solution(7) :-
Claim :- m∠1 and m∠2 are corresponding angles on two parallel lines with a transversal , which means their value will be equal .
Which means :-
[tex]4a + 67 = 7a + 34[/tex]
[tex]34 = 4a + 67 - 7a[/tex]
[tex]34 = - 3a + 67[/tex]
[tex] - 3a + 67 = 34[/tex]
[tex] - 3a = 34 - 67[/tex]
[tex] - 3a = - 33[/tex]
[tex]a = \frac{ - 33}{ - 3} [/tex]
[tex]\mapsto \: a = 11[/tex]
[tex]\color{olive}\implies\color{hotpink} x \color{hotpink}= 11[/tex]
Now , we know that the value of x is 11 , let us use this to find the measure of other angles :-
m∠1 = 111°
[tex] = 4a + 67[/tex]
[tex] = 4 \times 11 + 67[/tex]
[tex] = 44 + 67[/tex]
[tex]\color{olive}\implies\color{hotpink}m∠1 = 111°[/tex]
m∠2 = 111°
[tex] = 7a + 34[/tex]
[tex] = 7 \times 11 + 34[/tex]
[tex] = 77 + 34[/tex]
[tex]\color{olive}\implies \:\color{hotpink} m∠2 = 111°[/tex]
[tex]\color{teal}\therefore \: x = 11 \: , \: m∠1 = 111° \: , \: m∠2 = 111°[/tex]
Solution (8) :-
Claim :- m∠4 and m∠6 are interior angles on the same side of transversal , which means their sum will be equal to 180° .
Which means :-
[tex]3x+31 + 8x-5 = 180[/tex]
[tex]8x + 3x + 31 - 5 = 180[/tex]
[tex]11x + 31 - 5 = 180[/tex]
[tex]11 x + 26= 180[/tex]
[tex]11x = 180 - 26[/tex]
[tex]11x = 154[/tex]
[tex]x = \frac{154}{11} [/tex]
[tex]\color{olive}\implies\color{hotpink}x = 14[/tex]
m∠4 = 73°
[tex] = 3x + 31[/tex]
[tex] = 3 \times 14+ 31[/tex]
[tex] = 42 + 31[/tex]
[tex]\color{olive}\implies\color{hotpink} m∠4 = 73°[/tex]
m∠6 = 107°
[tex] = 8x - 5[/tex]
[tex] = 8 \times 14 - 5[/tex]
[tex] = 112 - 5[/tex]
[tex]\color{olive}\implies\color{hotpink}m∠6= 107°[/tex]
[tex]\color{teal}\therefore \: x = 14 \: , \: m∠4 = 63° , \: m∠6 = 107° [/tex]
