Question

118 H. M. Mathematics (Class 9)
10. In fig. find Zx.
D
с
(x + 20°)
(x + 31)
E
А
B
11. In the given fig. AB and AC are opposite rays. If (a -36) = 20°, find Za and 2
D
с
A
B
12. In the given fig. determine the value of x'.
2x
2x
60° /​

Answer 1

Given:-

• $\angle$ DAE = (x° + 31°)
• $\angle$ DAC = (x° + 20°)
• $\angle$ CAB = x°

Find:-

• Value of $\angle$ x

Diagram:-

$\setlength{\unitlength}{1 cm}\begin{picture}(0,0)\thicklines\qbezier(1,1)(1,1)(6,1)\qbezier(3,1)(6,4)(6,4)\qbezier(3,1)(1.5,4)(1.5,4)\put(3, 1){\vector(1,0){3}} \put(3,1){\vector( - 1,0){3}}\put(3,1){\vector(-1,2){2}}\put(3,1){\vector(1,1){3}} </p><p>\put(0.5,1){\circle*{0.3}}\put(5.5,1){\circle*{0.3}}\put(1.5,4){\circle*{0.3}}\put(5.5,3.5){\circle*{0.3}}\put(0.4,1.5){ \bf (x^{\circ} + 31^{\circ}) } \put(2.4,2){ \footnotesize{ \sf{(x^{\circ}+20^{\circ})}} }\put(4,1.4){ \bf (x^{ \circ}) } \qbezier(2.7,1.5)(1.9,1.5)(2,1)\qbezier(2.7,1.5)(3,2)(3.5,1.5) \qbezier(3.5,1.5)(4.4,1.4)(3.8,1)\put(0.5,0.5){\bf E} \put(2.8,0.6){\bf A} \put(5.4,0.5){\bf B}\put(1,3.8){\bf D}\put(5.7,3.2){\bf C} \end{picture}$

Solution:-

Here, we can see EAB is a straight line. So, sum of all angles lies on it will be 180° as because of Linear Pair.

$\implies\sf \angle DAE + \angle DAC + \angle CAB = {180}^{ \circ} \bigg\lgroup Linear\:Pair\bigg\rgroup$

$\purple{\sf where} \footnotesize{\begin{cases} \pink{\sf\angle DAE = ( {x}^{ \circ} + {31}^{ \circ})} \\ \green{\sf\angle DAC= ( {x}^{ \circ} + {20}^{ \circ})} \\ \blue{\sf \angle CAB = x^{\circ}} \end{cases}}$

☯$\underline{\textsf{Substituting these values in the formula:-}}$

$\dashrightarrow\sf \angle DAE + \angle DAC + \angle CAB = {180}^{ \circ}$

$\dashrightarrow\sf x^{\circ} + 31 ^{\circ}+ x ^{\circ} + 20 ^{\circ} +x^{\circ} = {180}^{ \circ}$

$\dashrightarrow\sf 3x^{\circ} + 51^{\circ}= {180}^{ \circ}$

$\dashrightarrow\sf 3x^{\circ}= {180}^{ \circ} - 51^{\circ}$

$\dashrightarrow\sf 3x^{\circ}= {129}^{ \circ}$

$\dashrightarrow\sf x = \dfrac{129}{3}$

$\dashrightarrow\sf x = 43$

$\dashrightarrow\sf x = 43^{ \circ}$

$\small{\sf \therefore \underline{Value\:of\:\angle x\:is\:43^{\circ}}}$