Question

In an Isosceles Triangle, the vertex angles is twice of either of the base angles. Find the angles of triangles

Answer 1

Answer:

90° , 45°,45°

Step-by-step explanation:

given, in the isosceles triangle, if AB = BC, we have ∠A = 2∠B = 2∠C  

(∠ B = ∠C as they are the base angles)

if the value of ∠B = x, then we have ∠C = x and ∠A = 2∠B = 2x

so, the angles are 2x, x,x

=> 2x + x+ x = 180°

=> 4x = 180°

=> x = 180°/4

=> x = 45°

so, ∠A = 2(45°) = 90°

     ∠B = 45°

     ∠C = 45°

Answer 2

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

Let us consider an isosceles ∆ ABC, with AB = AC.

Since,

AB = AC

⇛ ∠ABC = ∠ACB = 'x' (say) -----(1)

[Angle opposite to equal sides are always equal]

According to statement,

The vertex angles is twice of either of the base angles.

⇛ ∠BAC = 2∠ABC = 2x

Now,

We know,

Sum of all angles of a triangle is 180°.

Therefore,

∠ABC + ∠BCA + ∠BAC = 180°

⇛ x + x + 2x = 180°

⇛ 4x = 180°

⇛ x = 45°

Hence,

$\begin{gathered}\begin{gathered}\bf \:Angles \: of \: \triangle \:ABC \: are \: \begin{cases} &\sf{ \angle \:ABC \: = x = 45 \degree \: } \\ &\sf{\angle \:BCA = x = 45\degree \:}\\ &\sf{\angle \:BAC = 2x = 90\degree \:} \end{cases}\end{gathered}\end{gathered}$

Additional Information :-

Properties of a triangle

• A triangle has three sides, three angles, and three vertices.

• The sum of all internal angles of a triangle is always equal to 180°. This is called the angle sum property of a triangle.

• The sum of the length of any two sides of a triangle is greater than the length of the third side.

• The side opposite to the largest angle of a triangle is the largest side.

• Any exterior angle of the triangle is equal to the sum of its interior opposite angles. This is called the exterior angle property of a triangle.

Based on the angle measurement, there are three types of triangles:

• Acute Angled Triangle : A triangle that has all three angles less than 90° is an acute angle triangle.

• Right-Angled Triangle : A triangle that has one angle that measures exactly 90° is a right-angle triangle.

• Obtuse Angled Triangle : triangle that has one angle that measures more than 90° is an obtuse angle triangle.