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Answer 1

★ Concept :-

Here the concept of Theorem of Angle of Triangle has been used. We see that we are given a triangle whose three sides are equal to each other. In order to solve these type of problems easily, we shall firstly name the different points. Then using the different properties of triangles and applying values, we shall find the answer.

Let's do it !!

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★ Formula Used :-

$\\\;\boxed{\sf{\pink{Sum\;of\;all\;angles\;of\;Delta\;=\;\bf{180^{\circ}}}}}$

$\\\;\boxed{\sf{\pink{Sum\;of\;Linear\;Pair\;Angles\;=\;\bf{180^{\circ}}}}}$

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★ Solution :-

*Note :: The figure in the attachment has been named and that naming is used to find the solution.

Given,

» ∠ABC = 80°

» ∠ACB = x

» ∠CBA = y

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~ For properties and relations between different angles of triangle ::

✒ Angle Sum Property of ∆ : This states that sum of all angles of triangle equal to 180° .

✒ Linear Pair Property : This states if angles lie on a straight line, then sum of all angles which lie on the line equal to 180° .

✒ Opposite Side Property : This states that if opposites sides are equal, then the angle formed by these are also equal.

• In triangle ABC -

By angle sum property,

→ ∠BAC + ∠ACB + ∠ABC = 180°

Since, AB = BC

→ ∠BAC = ∠ACB = x

• In triangle BCD -

By angle sum property,

→ ∠BCD + ∠CBD + ∠CDB = 180°

Since, BC = CD

→ ∠CDB = ∠CBD = y

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~ For the value of x ::

By angle sum property, we know that

$\\\;\sf{\rightarrow\;\;Sum\;of\;all\;angles\;of\;Delta\;=\;\bf{180^{\circ}}}$

By applying value in this for ∆ABC, we get

$\\\;\sf{\rightarrow\;\;\angle BAC\;+\;\angle ACB\;+\;\angle ABC\;=\;\bf{180^{\circ}}}$

$\\\;\sf{\rightarrow\;\;x\;+\;x\;+\;80^{\circ}\;=\;\bf{180^{\circ}}}$

Since, ∠BAC = ∠ACB = x

$\\\;\sf{\rightarrow\;\;2x\;=\;\bf{180^{\circ}\;-\;80^{\circ}}}$

$\\\;\sf{\rightarrow\;\;2x\;=\;\bf{100^{\circ}}}$

$\\\;\sf{\rightarrow\;\;x\;=\;\bf{\dfrac{100^{\circ}}{2}}}$

$\\\;\bf{\rightarrow\;\;x\;=\;\bf{\blue{50^{\circ}}}}$

This is the answer.

$\\\;\underline{\boxed{\tt{Hence,\;\:value\;\:of\;\:x\;=\;\bf{\purple{50^{\circ}}}}}}$

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~ For the value of y ::

For this, firstly we need to find the value of ∠BCD.

We know that,

$\\\;\sf{\rightarrow\;\;Sum\;of\;Linear\;Pair\;Angles\;=\;\bf{180^{\circ}}}$

Since x and ∠BCD are in Linear Pair. So,

$\\\;\sf{\rightarrow\;\;x\;+\;\angle BCD\;=\;\bf{180^{\circ}}}$

$\\\;\sf{\rightarrow\;\;50^{\circ}\;+\;\angle BCD\;=\;\bf{180^{\circ}}}$

$\\\;\sf{\rightarrow\;\;\angle BCD\;=\;\bf{180^{\circ}\;-\;50^{\circ}}}$

$\\\;\bf{\rightarrow\;\;\angle BCD\;=\;\bf{\red{130^{\circ}}}}$

Now by applying Angle Sum Property in ∆BCD, we get

$\\\;\sf{\rightarrow\;\;\angle BCD\;+\;\angle CDB\;+\;\angle CBD\;=\;\bf{180^{\circ}}}$

$\\\;\sf{\rightarrow\;\;130^{\circ}\;+\;y\;+\;y\;=\;\bf{180^{\circ}}}$

Since, ∠CBD = ∠CDB = y

$\\\;\sf{\rightarrow\;\;2y\;=\;\bf{180^{\circ}\;-\;1300^{\circ}}}$

$\\\;\sf{\rightarrow\;\;2y\;=\;\bf{50^{\circ}}}$

$\\\;\sf{\rightarrow\;\;y\;=\;\bf{\dfrac{50^{\circ}}{2}}}$

$\\\;\bf{\rightarrow\;\;y\;=\;\bf{\blue{25^{\circ}}}}$

This is the answer.

$\\\;\underline{\boxed{\tt{Hence,\;\:value\;\:of\;\:y\;=\;\bf{\green{25^{\circ}}}}}}$

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★ More to know :-

• Equilateral Triangle : Triangle whose all the sides are equal and all the angles equal to 60° .

• Right Triangle : Triangle whose one angle is 90° is known as Right Triangle.

Answer 2

Given

• ☆ ∠ABC = 80°
• ☆ ∠ACB = x
• ☆ ∠CBD = y

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To Find :-

• ☆ The value of x and y.

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Concept Used:-

• Angle opposite to equal sides are always equal.
• Sum of angles of a triangle is supplementary.
• Linear pair of angles is supplementary.
• Exterior angle of a triangle is always equal to sum of interior opposite angles.

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Solution :-

☆ In triangle ABC -

☆ Since, AB = BC

⇛ ∠BAC = ∠ACB = x

☆ By angle sum property,

☆ ∠BCA + ∠ACB + ∠ABC = 180°

⇛ 80° + x + x = 180°

⇛ 80° + 2x = 180°

⇛2x = 180° - 80 °

⇛ 2x = 100°

⇛ x = 50°.

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☆ In triangle BCD -

☆ Since, BC = CD

⇛ ∠CDB = ∠CBD = y

☆ By exterior angle property,

⇛ ∠BCA = ∠CBD + ∠CDB

⇛ 50° = y + y

⇛ 2y = 50°

⇛ y = 25°

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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.