Question

In the figure find the value of x and y​

Answer 1

★ Understanding the question :

» Here we are given a figure in which two triangles (∆ABC , ∆DBE) are joined together.

» In ∆ABC ∠A = 50° , ∠C = 30° , ∠B(3) in not given and in ∆DBE ∠D = 55° , measure of ∠B(1) and ∠E(4),(2) is not given.

» We had to find out the values of x° and y°.

Let's do it !!

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$\underbrace{\underline{\sf{Required\:solution\::}}}$

★ Using angle sum property on ∆ABC :

$\qquad:\implies\:\sf \angle{A} + \angle{3} + \angle{C} = 180\degree$

★ Putting value of ∠A and ∠C :

$\qquad:\implies\:\sf 50\degree + \angle{3} + 30\degree = 180\degree$

$\qquad:\implies\:\sf \angle{3} + 80\degree = 180\degree$

$\qquad:\implies\:\sf \angle{3} = 180\degree - 80\degree$

$\qquad:\implies\:\bold \angle{3} = \red{100\degree}$

Now,

$\qquad:\rightarrow\:\sf \angle{3} + \angle{1} = 180\degree \:\:\lgroup Linear\:pair \rgroup$

★ Putting value of ∠3 :

$\qquad:\rightarrow\:\sf 100\degree + \angle{1} = 180\degree$

$\qquad:\rightarrow\:\sf \angle{1} = 180\degree - 100\degree$

$\qquad:\rightarrow\:\bold \angle{1} = \red{80\degree}$

★ Using angle sum property on ∆DBE :

$\qquad:\implies\:\sf \angle{D} + \angle{1} + \angle{4} = 180\degree$

★ Putting value of ∠D and ∠1 :

$\qquad:\implies\:\sf 55\degree + 80\degree + \angle{4} = 180\degree$

$\qquad:\implies\:\sf 135\degree + \angle{4} = 180\degree$

$\qquad:\implies\:\sf \angle{4} = 180\degree - 135\degree$

$\qquad:\implies\:\bold \angle{4} = \red{45\degree}$

Now,

$\qquad:\rightarrow\:\sf \angle{4} + \angle{2} = 180\degree \:\:\lgroup Linear\:pair \rgroup$

★ Putting value of ∠4 :

$\qquad:\rightarrow\:\sf 45\degree + \angle{2} = 180\degree$

$\qquad:\rightarrow\:\sf \angle{2} = 180\degree - 45\degree$

$\qquad:\rightarrow\:\bold \angle{2} = \red{135\degree}$

$\underline{\boxed{\tt{Values\:of\:x\degree\:and\:y\degree\:=\:\rm\purple{135\degree}\:and\:\rm\purple{80\degree}}}}$

Answer 2

Given:

• There is two triangle first one is ∆ABC and second one is ∆BDE.
• ∠CAB = 50°
• ∠ACB is 30°
• ∠CED = x°
• ∠EBD = y°
• ∠BDE = 55°

To Find:

• Value of x and y.

Solution:

In triangle 1st :

• ∠CAB = 50°
• ∠ACB = 30°

Therefore,

∠CAB + ∠ACB + ∠ABC = 180° •••••(Angle sum property)

→ 50° + 30° + ∠ABC = 180°

→ 80° + ∠ABC = 180°

→ ∠ABC = 180° - 80°

→ ∠ABC = 100°

Now, in ∆ABC :

• ∠CAB = 50°
• ∠ACB = 30°
• ∠ABC = 100°

Now,

∠ABC + ∠EBD = 180° ••••••(linear pair)

→ 100° + ∠EBD = 180°

→ ∠EBD = 180° - 100°

→ ∠EBD = 80°

Therefore,

• $\large{\underline{\boxed{\bf{\red{y~=~80°}}}}}$

_______________

In triangle 2nd :

• ∠EBD = 80°
• ∠BDE = 55°

Therefore,

∠BED +∠EBD +∠BDE = 180° •••••(Angle sum property)

→ ∠BED+ 80° + 55° = 180°

→ ∠BED + 135° = 180°

→ ∠BED= 180° - 135°

→ ∠BED = 45°

Therefore,

∠EBD + ∠BDE = ∠CED •••••(Exterior angle property)

→ 80° + 55° = x

→ 135° = x

Therefore,

• $\large{\underline{\boxed{\bf{\red{x~=~135°}}}}}$

Know more :

• Angle sum property of traingle state that sum of interior angle of traingle is equal to 180°
• Linear pair is a pair of adjacent angle formed when two line intersect.
• Exterior angle property state that exterior angle of traingle is equal to two interior opposite angles of triangle.