1 The sides BC, CA and AB of AABC are produced in order to form exterior angles ZACD, ZBAE and ZCBF. Show that ZACD + ZBAE + ZCBF = 360°.
Answers
Step-by-step explanation:
Given :-
The sides BC, CA and AB of AABC are produced in order to form exterior angles ∠ACD, ∠BAE and ∠ CBF.
Required To Prove :-
Show that ∠ACD +∠BAE +∠CBF = 360°.
Proof-1:- :-
Given that
∆ ABC is a triangle with the sides AB, BC and CA
The three angles are ∠BAC , ∠ABC and ∠CBA
We know that
The sum of all the three angles in a triangle is 180°.
=> ∠BAC +∠ABC +∠CBA = 180° ------(1)
The side BC produced to D then
∠BCA + ∠ACD = 180° --------(2)
(linear pair)
The side CA is produced to E then
∠CAB + ∠BAE = 180° --------(3)
(linear pair)
and The side AB is produced to F then
∠CBF+ ∠CBA = 180° --------(4)
On adding (2),(3)&(4)
=> ∠BCA + ∠ACD+∠CAB + ∠BAE+∠CBF+ ∠CBA = 180°+180°+180°
=> ∠BCA + ∠ACD+∠CAB + ∠BAE+∠CBF+ ∠CBA = 540°
=> (∠BAC +∠ABC +∠CBA )+(∠ACD +∠BAE +∠CBF )
= 540°
=> 180°+∠ACD +∠BAE +∠CBF = 540°
=> ∠ACD +∠BAE +∠CBF = 540°-180°
=> ∠ACD +∠BAE +∠CBF = 360°
Hence , Proved.
Proof-2:-
Given that
∆ ABC is a triangle with the sides AB, BC and CA
The three angles are ∠BAC , ∠ABC and ∠CBA
We know that
The sum of all the three angles in a triangle is 180°.
=> ∠BAC +∠ABC +∠CBA = 180° ------(1)
The side BC produced to D then ∠ACD is an exterior angle
We know that
The exterior angle is equal to the sum of two opposite interior angles
=> ∠ACD = ∠BAC +∠CBA-------(2)
The side CA is produced to E then ∠BAE is qn exterior angle
We know that
The exterior angle is equal to the sum of two opposite interior angles
∠BAE = ∠CBA +∠ACB --------(3)
and The side AB is produced to F then ∠CBF is an exterior angle
We know that
The exterior angle is equal to the sum of two opposite interior angles
∠CBF= ∠BCA +∠BAC --------(4)
On adding (2),(3)&(4)
=> ∠ACD +∠BAE +∠CBF
=> ∠BAC +∠CBA+∠CBA +∠ACB+∠BCA +∠BAC
=>∠BAC +∠BAC +∠ABC +∠ABC +∠CBA+∠CBA
=> 2(∠BAC +∠ABC +∠CBA )
=> 2(180°) (frim (1))
=> 360°
∠ACD +∠BAE +∠CBF = 360°
Hence, Proved.
" The three sides of a triangle are extended formed three exterior angles then the sum of the three exterior angles is equal to 360°".
Used formulae:-
Angle Sum Property:-
→ The sum of all the three angles in a triangle is 180°.
Linear pair:-
→ The sum of two adjacent angles is equal to 180° is called Linear Pair.
Exterior angle property:-
→ The exterior angle is equal to the sum of two opposite interior angles in a triangle.
Step-by-step explanation:
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