Question

find the values of x y z in the following figure​​

Answer 1

Answer:

x+y= 180° (liner pairs)______(1)

similarly y+z =180°_____(2)

& z+x/2=180°_____(3)

Step-by-step explanation:

from equation 1

x=z(vertically opposite number)

from equation 3

Answer 2

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°