Question

LMNP is a cyclic quadrilateral seg LM is extended upto point T if m angle LPN= 105° then angle NMT =?

Plz explain STEP BY STEP plz​

Answer 1

Given:-

• Given quadrilateral is cyclic quadrilateral.
• ∠LPN is 105°.

To find:-

• ∠NMT.

Solution:-

We know that,

Sum of either pair of opposite angles of a cyclic quadrilateral is 180°.

So,

⇒∠LPN + ∠LMN = 180°

⇒105° + ∠LMN = 180°

⇒∠LMN = 180° - 105°

⇒∠LMN = 75°

We also know that,

Sum of all angles forms on straight line is equal to 180°. This statement is also known as linear pair.

So,

⇒∠LMN + ∠NMT = 180°[Linear pair]

⇒75° + ∠NMT = 180°

⇒∠NMT = 180° - 75°

⇒∠NMT = 105°

Therefore,

∠NMT is 105°.

Answer 2

$\pink{\large{\underline{\underline{ \rm{Given: }}}}}$

◉ LMNP is a cyclic quadrilateral.

◉ LM is extended upto point T.

◉ ∠LPN = 105°

$\pink{\large{\underline{\underline{ \rm{To \: Find: }}}}}$

✿ ∠NMT.

$\pink{\large{\underline{\underline{ \rm{Solution: }}}}}$

A quadrilateral is cyclic when a circle passes through all its four vertices.

Also, opposite angles of a cyclic quadrilateral add up to 180°.

We have,

➩ ∠LPN + ∠LMN = 180°

➩ 105° + ∠LMN = 180°

➩ ∠LMN = 180° - 105°

➩ ∠LMN = 75°

A pair of adjacent angles is called linear pair if they add up to 180°. So here ∠LMN and ∠NMT is a pair of adjacent angles and if we add them then we get their measure 180°, therefore it is a linear pair.

So, we have:

➩ ∠LMN + ∠NMT = 180°

➩ ∠75° + ∠NMT = 180°

➩ ∠NMT = 180° - 75°

➩ ∠NMT = 105°

∴ ∠NMT = $\green{ \underline{ \boxed{ \sf{105 \degree}}}}$

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