Question

If a ray stands on a line, then the sum of the adjacent angles so formed is 180°.​

Answer 1

$\huge{\underline{\mathfrak{Solution:}}}$

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$\sf{\large{\underline{\underline{Given \colon}}}}$

A ray CD which stands on a line AB such that $\sf\angle ACD$ and $\sf \angle BCD \: are \: formed.$

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$\sf{\large{\underline{\underline{To \: Prove \colon}}}}$

$\sf \angle ACD + \angle BCD = 180 \degree.$

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$\sf{\large{\underline{\underline{Construction \colon}}}}$

Draw ray $\sf CE \perp AB.$

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$\sf{\large{\underline{\underline{Prove \colon}}}}$

We have,

$\angle$ACD = $\angle$ACE + $\angle$ECD .....(i)

and,

$\angle$BCD = $\angle$BCE $- \angle$ECD .....(ii)

Adding (i) and (ii), we get

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf \angle ACD + \angle BCD = \angle A C E + \angle ECD + \angle BCE - \angle ECD \\ \sf \implies \: \: \: \: \: \: \: \angle ACD + \angle BCD = \angle A C E + \angle BCE \\ \sf \implies \: \: \: \: \: \: \: \angle ACD + \angle BCD = 90 \degree + 90 \degree \\ \sf \implies \: \: \: \: \: \: \: \angle ACD + \angle BCD = 180 \degree \\ \\ \sf Hence , \huge\boxed { \boxed{ \blue{\small \sf\angle ACD + \angle BCD = 180 \degree}}}$

✔✔ Hence, it is proved ✅✅.

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Answer 2

ANSWER

When a ray stands on a line, two adjacent angles are formed.

We know that the angle lying on a straight line is 180°.

The two angles being adjacent, make a total angle of 180° on the straight line.

Another way, we can see since the ray stands on the straight line, we can consider it is a perpendicular line.

Thus, the two adjacent angles are right angles.

So, the total angle

= 90° + 90°

= 180°