Question

Find the unknown angle​

Answer 1

${\underline {\boxed {\Large {\bf \gray { {50}^{\circ} } }}}}$

Given Information:

• $\angle$ BAE = 110°

• $\angle$ ECD = 120°

To calculate:

• The value of x° i.e $\angle$ ABC.

★ Procedure:

Here, we will first calculate the measure of other two interior angles of the triangle. To calculate this we are given enough information, by applying the linear pair property we can find the measure of two interior angles of the given triangle.

Then, by angle sum property of the triangle, we'll calculate the value of x°.

Calculations:

As, we know that :

• The sum of the angles lie on the straight line is 180°. Here, EC and BD are two straight lines.

$\sf{\longrightarrow \: \angle BAE + \angle BAC = {180}^{\circ} \: [Linear \: pair ] } \\ \\ \\ \sf{\longrightarrow \: \angle {110}^{\circ} + \angle A = {180}^{\circ} } \\ \\ \\ \sf{\longrightarrow \: \angle A = {180}^{\circ} - \angle {110}^{\circ}}\\ \\ \\ \longrightarrow \: \underline{\boxed{\sf{\angle A = {70}^{\circ}}}} \: \red{\bigstar}$

Also,

$\sf{\longrightarrow \: \angle ACD + \angle ACB = {180}^{\circ} \: [Linear \: pair ] } \\ \\ \\ \sf{\longrightarrow \: \angle {120}^{\circ} + \angle C = {180}^{\circ} } \\ \\ \\ \sf{\longrightarrow \: \angle C = {180}^{\circ} - \angle {120}^{\circ}}\\ \\ \\ \longrightarrow \: \underline{\boxed{\sf{\angle C = {60}^{\circ}}}} \: \red{\bigstar}$

Now, as we also know that :

• Sum of the measure of interior angles of a triangle is equivalent to 180°.

$\sf {\longrightarrow \angle A + \angle B + \angle C = {180}^{\circ}} \\ \\ \\ \sf {\longrightarrow {70}^{\circ} + {x}^{\circ} + {60}^{\circ} = {180}^{\circ}} \\ \\ \\ \sf {\longrightarrow {130}^{\circ} +{x}^{\circ} = {180}^{\circ}} \\ \\ \\ \sf {\longrightarrow {x}^{\circ} = {180}^{\circ}- {130}^{\circ}} \\ \\ \\ \underline{\boxed{\sf{ {x}^{\circ} = {50}^{\circ}}}} \: \red{\bigstar}$

Henceforth,

• Value of x° is 50°

Extra Information:

Important properties of triangle :

★ Angle sum property of a triangle :

• Sum of interior angles of a triangle = 180°

★ Exterior angle property of a triangle :

• Sum of two interior opposite angles = Exterior angle

★ Perimeter of triangle :

• Sum of all sides

★ Area of triangle :

• $\sf { \dfrac{1}{2} \times Base \times Height }$

★ Area of an equilateral triangle:

• $\sf { \dfrac{\sqrt{3}}{4} \times {Side}^{2} }$

★ Area of a triangle when its sides are given :

• $\sf { \sqrt{s[ (s-a)(s-b)(s-c) ]} }$

Where,

• S= Semi-perimeter or $\sf {\dfrac{a+b+c}{2} }$

Answer 2

In △ABC, ∠ABC + ∠BAC = ∠ACD (exterior angle property)

⇒ x + ∠BAC = 120°

⇒ x + (180° - ∠BAE) = 120° (by linear pair)

⇒ x + 70° = 120°

⇒ x = 120° - 70°

⇒ x = 50°.

∴ The unknown angle, i.e., ∠ABC or x is equal to 50°.