find X and y in the figure
Answers
Step-by-step explanation:
Given :-
In ∆ABC, BC , CA and AB is produced to X, Y and Z .
∠XCA = 120°
∠YAZ = 60°
To find :-
i) The angles of ∆ABC
ii) Type of ∆ABC
Solution :-
Method-1:-
Given that
In ∆ABC, BC , CA and AB is produced to X, Y and Z .
∠XCA = 120°
∠XCA = 120°∠YAZ = 60°
From the given figure,
∠YAZ and ∠CAB are vertically opposite angles.
We know that
Vertically opposite angles are equal.
Therefore, ∠YAZ = ∠CAB
=> ∠CAB = 60°
and
∠XCA and ∠ BCA are linear Pair
Therefore, ∠ XCA + ∠ BCA = 180°
=> 120°+∠BCA = 180°
=> ∠ BCA = 180°-120°
Therefore, ∠ BCA = 60°
We know that
Interior angles sum property of a triangle.
∠CAB+∠ABC+∠BCA = 180°
=> 60° + ∠ ABC + 60° = 180°
=> ∠ABC + 120° = 180°
=> ∠ABC = 180° -120°
=> ∠ ABC = 60°
Therefore, ∠ ABC = 60°
We have,
∠CAB = 60°
∠ ABC = 60°
∠ BCA = 60°
The angles of the triangle ABC are 60°, 60° and 60°
So ,all angles are equal.
Therefore, ∆ABC is an equilateral triangle.
Method-2:-
Given that
In ∆ABC, BC , CA and AB is produced to X, Y and Z .
∠XCA = 120°
∠YAZ = 60°
From the given figure,
∠YAZ and ∠ CAB are vertically opposite angles.
We know that
Vertically opposite angles are equal.
Therefore, ∠ YAZ = ∠CAB
=> ∠CAB = 60°
And ∠ XCA and ∠ BCA are linear pair
Therefore, ∠ XCA + ∠BCA = 180°
=> 120°+∠BCA = 180°
=> ∠BCA = 180°-120°
Therefore, ∠ BCA = 60°
We know that
Interior angles sum property of a triangle
∠XCA is an exterior angle formed by extending BC to D.
We know that
An exterior angle is equal to the sum of its opposite interior angles in a triangle.
=> ∠ XCA = ∠CAB+ ∠ABC
=> 120° = 60° + ∠ABC
=> ∠ ABC = 120°-60°
=> ∠ABC = 60°
We have,
∠CAB = 60°
∠CAB = 60°∠ ABC = 60°
∠CAB = 60°∠ ABC = 60°∠ BCA = 60°
The angles of ∆ABC are 60° , 60° and 60°
So ,all angles are equal.
Therefore, ∆ABC is an equilateral triangle.
Answer :-
i) The angles of the ∆ABC are 60° , 60° and 60°
ii) ∆ABC is an equilateral triangle.
Used formulae:-
→ Vertically opposite angles are equal.
Interior angles sum property:-
" The sum of all the three interior angles of a triangle is equal to 180°".
Equilateral triangle:-
" A triangle with all the three sides are equal is called an equilateral triangle.
→ All angles are equal.
→ Each angle is 60°
Exterior angle of a triangle :-
→ An exterior angle is equal to the sum of its opposite interior angles in a triangle.
Linear Pair :-
The sum of two adjacent angles is 180° are called a Linear Pair.
Step-by-step explanation:
Step-by-step explanation:
Given :-
In ∆ABC, BC , CA and AB is produced to X, Y and Z .
∠XCA = 120°
∠YAZ = 60°
To find :-
i) The angles of ∆ABC
ii) Type of ∆ABC
Solution :-
Method-1:-
Given that
In ∆ABC, BC , CA and AB is produced to X, Y and Z .
∠XCA = 120°
∠XCA = 120°∠YAZ = 60°
From the given figure,
∠YAZ and ∠CAB are vertically opposite angles.
We know that
Vertically opposite angles are equal.
Therefore, ∠YAZ = ∠CAB
=> ∠CAB = 60°
and
∠XCA and ∠ BCA are linear Pair
Therefore, ∠ XCA + ∠ BCA = 180°
=> 120°+∠BCA = 180°
=> ∠ BCA = 180°-120°
Therefore, ∠ BCA = 60°
We know that
Interior angles sum property of a triangle.
∠CAB+∠ABC+∠BCA = 180°
=> 60° + ∠ ABC + 60° = 180°
=> ∠ABC + 120° = 180°
=> ∠ABC = 180° -120°
=> ∠ ABC = 60°
Therefore, ∠ ABC = 60°
We have,
∠CAB = 60°
∠ ABC = 60°
∠ BCA = 60°
The angles of the triangle ABC are 60°, 60° and 60°
So ,all angles are equal.
Therefore, ∆ABC is an equilateral triangle.
Method-2:-
Given that
In ∆ABC, BC , CA and AB is produced to X, Y and Z .
∠XCA = 120°
∠YAZ = 60°
From the given figure,
∠YAZ and ∠ CAB are vertically opposite angles.
We know that
Vertically opposite angles are equal.
Therefore, ∠ YAZ = ∠CAB
=> ∠CAB = 60°
And ∠ XCA and ∠ BCA are linear pair
Therefore, ∠ XCA + ∠BCA = 180°
=> 120°+∠BCA = 180°
=> ∠BCA = 180°-120°
Therefore, ∠ BCA = 60°
We know that
Interior angles sum property of a triangle
∠XCA is an exterior angle formed by extending BC to D.
We know that
An exterior angle is equal to the sum of its opposite interior angles in a triangle.
=> ∠ XCA = ∠CAB+ ∠ABC
=> 120° = 60° + ∠ABC
=> ∠ ABC = 120°-60°
=> ∠ABC = 60°
We have,
∠CAB = 60°
∠CAB = 60°∠ ABC = 60°
∠CAB = 60°∠ ABC = 60°∠ BCA = 60°
The angles of ∆ABC are 60° , 60° and 60°
So ,all angles are equal.
Therefore, ∆ABC is an equilateral triangle.
Answer :-
i) The angles of the ∆ABC are 60° , 60° and 60°
ii) ∆ABC is an equilateral triangle.
Used formulae:-
→ Vertically opposite angles are equal.
Interior angles sum property:-
" The sum of all the three interior angles of a triangle is equal to 180°".
Equilateral triangle:-
" A triangle with all the three sides are equal is called an equilateral triangle.
→ All angles are equal.
→ Each angle is 60°
Exterior angle of a triangle :-
→ An exterior angle is equal to the sum of its opposite interior angles in a triangle.
Linear Pair :-
The sum of two adjacent angles is 180° are called a Linear Pair.