Find the values of x y z in the following figure

1.	Find the values of x y z in the following figure
Answer» Step-by-step explanation: As per the figure, ⠀⠀⠀★ ∠1 = x ⠀⠀⠀★ ∠2 = x/2 ⠀⠀⠀★ ∠3 = z ⠀⠀⠀★ ∠4 = y We are asked to calculate the value of x,y,z. Clearly, ∠1 and ∠3 ; ∠2 and ∠4 are vertically opposite angles. Since, the vertically opposite angles equal, so ∠1 and ∠3 ; ∠2 and ∠4 will be equal. Thus, we can say that : ➝ ∠1 = ∠3 ⇒ x ➝ ∠2 = ∠4 ⇒ x/2 Value of y can be written as x/2. Value of z can be written as x. The sum of all these angles will be 360° as they are forming a complete angle. WRITING it in the FORM of a linear EQUATION, $\longrightarrow \sf{\quad {\angle 1 + \angle 2 + \angle 3 + \angle 4 = 360^\circ }} \\$ Substitute the measure angles. $\longrightarrow \sf{\quad {x+ \dfrac{x}{2} + z + y = 360^\circ }} \\$ Now, substitute the expression of y and z which have been found using the property of vertically opposite angles. $\longrightarrow \sf{\quad {x+ \dfrac{x}{2} + x + \dfrac{x}{2} = 360^\circ }} \\$ Taking the LCM and solving further. $\longrightarrow \sf{\quad {\dfrac{2x + x + 2x + x}{2} = 360^\circ }} \\$ Performing addition in the numerator of the FRACTION in the LHS. $\longrightarrow \sf{\quad {\dfrac{6x}{2} = 360^\circ }} \\$ Transposing 2 from L.H.S. to R.H.S. $\longrightarrow \sf{\quad {6x= 360^\circ \times 2 }} \\$ Performing multiplication in RHS. $\longrightarrow \sf{\quad {6x= 720^\circ }} \\$ Transposing 6 from L.H.S. to R.H.S. $\longrightarrow \sf{\quad {x= \cancel{ \dfrac{720^\circ}{6}} }} \\$ Dividing 720 by 6. $\longrightarrow \quad\underline{\boxed { \textbf{\textsf{x = 120}}^\circ }}\\$ Now, we have to FIND the value of other three angles. Value of ∠2 : x/2 = 120°/2 = 60° $\longrightarrow \quad\underline{\boxed { \angle\textbf{\textsf{2 = 60}}^\circ }}\\$ Value of ∠3 : Same as the value of x, since the vertically opposite angles are equal. $\longrightarrow \quad\underline{\boxed { \angle\textbf{\textsf{3 = 120}}^\circ }}\\$ Value of ∠4 : Same as the value of x/2, since the vertically opposite angles are equal. $\longrightarrow \quad\underline{\boxed { \angle\textbf{\textsf{4 = 60}}^\circ }}\\$ $\underline{ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad} \\$ Therefore, ⠀⠀⠀★ ∠1 = 120° ⠀⠀⠀★ ∠2 = 60° ⠀⠀⠀★ ∠3 = 120° ⠀⠀⠀★ ∠4 = 60°

Answer»

Step-by-step explanation:

As per the figure,

⠀⠀⠀★ ∠1 = x

⠀⠀⠀★ ∠2 = x/2

⠀⠀⠀★ ∠3 = z

⠀⠀⠀★ ∠4 = y

We are asked to calculate the value of x,y,z. Clearly, ∠1 and ∠3 ; ∠2 and ∠4 are vertically opposite angles. Since, the vertically opposite angles equal, so ∠1 and ∠3 ; ∠2 and ∠4 will be equal.

Thus, we can say that :

➝ ∠1 = ∠3 ⇒ x

➝ ∠2 = ∠4 ⇒ x/2

Value of y can be written as x/2.
Value of z can be written as x.

The sum of all these angles will be 360° as they are forming a complete angle. WRITING it in the FORM of a linear EQUATION,