Question

$\purple{\boxed{Theorem 6.8}}$ if a side of a triangle is produced then the exterior angle so formed is equal to the sum of the two interior opposite angle.​

Answer 1

${\tt{Question}}\: \purple☟$

$\purple{\boxed{\tt{Theorem\: 6.8}}}$ if a side of a triangle is produced then the exterior angle so formed is equal to the sum of the two interior opposite angle.

▬▬▬▬▬▬▬▬▬▬▬▬▬▬

$\red&#9998{\tt{Answer}}\: \orange☟$

⠀⠀$&#9679\purple{\bf{See \:the \:attachment}}\red{⇑}$

⠀ $&#9679\orange{\bf{Question\: solved}}\: \green✔$

▬▬▬▬▬▬▬▬▬▬▬▬▬▬

All necessary formulas⤵️

▬▬▬▬▬▬▬▬▬▬▬▬▬▬

$\red{\tt{ Here}}$ $\pink{\boxed{&#10811=Triangle}}$

$\orange\star{\bf{\red{\underbrace{complementary \:angle}}}}\red\star$

The sum of 2 numbers$=90°$

example $a−b=90°$

how to find "a" if a is not mentioned

$\red{\underbrace{\bf{\orange{Given࿐}}}}$

$a= \: ?$

$b = 40$

$a+40=\:90°$

$a=90-40°$

$a=50°$

▬▬▬▬▬▬▬▬▬▬▬▬▬▬

$\pink\star{\bf{\purple{\underbrace{supplementary\: angle}}}}\red\star$

The sum of two numbers$= \:180°$

example $a+b=180°$

how to find "a" if a is not mentioned

Given

$a= \:?$

$b =\: 40$

$a+40=180°$

$a=180-40°$

$a=140°$

▬▬▬▬▬▬▬▬▬▬▬▬▬▬

$\orange\star{\bf{\green{\underbrace{Adjacent \:angle}}}}\red\star$

If there is a common ray between ${\bf&#x2220}a$ and ${\bf&#x2220}b$ so it is a adjacent angle.

▬▬▬▬▬▬▬▬▬▬▬▬▬▬

$\orange\star{\bf{\blue{\underbrace{Vertical\: opposite\: angle }}}}\red\star$

Vertical angles are pair angles formed when two lines intersect. Vertical angles are sometimes referred to as vertically opposite angles because the angles are opposite to each other.

▬▬▬▬▬▬▬▬▬▬▬▬▬▬

$\orange\star{\bf{\orange{\underbrace{lenear\: pair \:of\: angles}}}}\red\star$

Here ${\bf&#x2220}a$$+{\bf&#x2220}b$$=$180°.

▬▬▬▬▬▬▬▬▬▬▬▬▬▬

Answer 2

Statement : If side of a triangle is produced, then the exterior angles so formed is equal to the sum of the two interior opposite angles.

Given : A triangle ABC with interior angles x , y and z and exterior angle 'e'.

To prove : e = x + y

PROOF :

In the figure :

x + y + z = 180°... (i) [Angle sum property]

e + z = 180 ° ... (ii) [Linear pair]

On camparing equation ( i ) and ( ii )

x + y + z = e + z

Therefore,

x + y = e

Hence, it is proved.