Question

$\huge \purple\star \large \green\star \small \red \star \tiny \orange\star \large{\sf{Solution}} \tiny \orange\star\small \red \star \large \green\star \huge \purple\star$
In the given figure, PQII RS, ZPXM = 50° and ZMYS = 120°, find the value of x if reflex angleXMY is x -20°.​

Answer 1

$\large\underline{\sf{Solution-}}$

Construction :- Through M, draw a line m parallel to PQ.

So, m || PQ || RS

As PQ || m

⟹ ∠PXM = ∠XMB [ Alternate interior angles ]

⟹∠XMB = 50° ------[1]

Now, m || RS

⟹ ∠BMY + ∠MYS = 180° [ Sum of co-interiors ]

⟹ ∠BMY + 120° = 180°

⟹ ∠BMY = 60° --------[2]

Now, ∠XMY = ∠XMB + ∠BMY

⟹ ∠XMY = 50° + 60° = 110°

Now,

∠XMY + reflex∠XMY = 360°

$\rm \: 110 \degree\: + \: x - 20 = 360\degree$

$\rm \: 90 \degree\: + \: x = 360\degree$

$\rm \: x = 360\degree - 90\degree$

$\rm\implies \:\boxed{ \rm{ \:x \: = \: 270\degree \: \: }}$

$\rule{190pt}{2pt}$

Additional Information :-

1. If lines are parallel, corresponding angles are equal.

2. If lines are parallel, alternate interior angles are equal.

3. If lines are parallel, sum of co-interior angles are supplementary.

Answer 2

$\huge \purple\star \large \green\star \small \red \star \tiny \orange\star \large{\sf{Solution}} \tiny \orange\star\small \red \star \large \green\star \huge \purple\star \\ \bf{ Draw \: a \: line through \: M \: Parallel \: to \: PQ \: and \: RS.} \\ \sf{ ∠ a=50⁰ (alternate angles)} \\ \sf{∠ c+120⁰ =180⁰ (linear pair)} \\ \sf{ = ∠ c=60⁰} \\ \sf{ Also \:∠b= ∠ c (alternate \: angles)} \\ \sf{=∠ b=60 }\\ \sf{x-20 + ∠b +∠ a=360 }\\ \sf{ = x-20⁰ +60⁰ +50⁰ =360⁰} \\\sf{x=360⁰-110⁰+20⁰}\\ \sf{=270⁰}$