Question

4. In Fig. 6.16, if x + y=w+z. then prove that AOB is a line.​

Answer 1

${\tt{Question}}\: \purple☟$

In Fig. 6.16, if x + y=w+z. then prove that AOB is a line.

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$\red&#9998{\tt{Answer}}\: \orange☟$

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

Sum of all angles in a circle always 360°

Hence

∠AOC + ∠BOC + ∠DOB + ∠AOD = 360°

=> x + y + w + z = 360°

=> x + y + x + y = 360°

Given that x + y = w + z

Plug the value we get

=> 2w + 2z = 360°

=> 2(w + z) = 360°

w + z = 180° (linear pair)

or ∠BOD + ∠AOD = 180°

If the sum of two adjacent angles is 180°, then the non-common arms of the angles form a line

Hence AOB is a line.

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

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All necessary formulas⤵️

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$\purple{\boxed{\bf{Angle\:sum\: property}}}$

Angle sum property of triangle states that the sum of interior angles of a triangle is 180°. Proof .Thus, the sum of the interior angles of a triangle is 180°.

$\blue{\tt{Example:-}}$

$\red{\boxed{a+b+c=180°}}$

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$\orange{\boxed{\bf{Alternate\:interior\:angle}}}$

Alternate interior angles are angles formed when two parallel or non-parallel lines are intersected by a transversal.Alternate interior angles are equal if the lines intersected by the transversal are parallel.

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

The sum of 2 numbers$=90°$

example $a−b=90°$

how to find "a" if a is not mentioned

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

$a= \: ?$

$b = 40$

$a+40=\:90°$

$a=90-40°$

$a=50°$

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

The sum of two numbers$= \:180°$

example $a+b=180°$

how to find "a" if a is not mentioned

Given

$a= \:?$

$b =\: 40$

$a+40=180°$

$a=180-40°$

$a=140°$

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

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

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

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.

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

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

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Answer 2

GIVEN

• x + y = w + z

TO PROVE

• AOB is a line

SOLUTION

• x + y + w + z = 360∘
• (x+y) + (w+z) = 360∘
• (x+y) + (x+y) = 360∘

Given x + y = w + z

• 2(x + y) = 360∘
• (x + y) = 180∘

Here, x + y is a linear pair.

Hence, AOB is a straight line