what is the counterexample for each conjecture? One and only one plane exists through any three points.

∠MLP = 72° ,           ∠LJK = 22° ,            ∠JKL = 72° ,          ∠KLJ  =  86°

Step-by-step explanation:

Here, given In ΔJLK and  ΔMLP

Here,  JK  II  ML,  LM = MP

∠JLM = 22° and  ∠LMP = 36°

Now, As angles opposite to equal sides are equal.

⇒ ∠MLP = ∠MPL  = x°

Now, in  ΔMLP

By ANGLE SUM PROPERTY:   ∠MLP + ∠MPL  + ∠LMP = 180°

⇒ x° + x° + 36° = 180°

⇒ 2 x  = 180 - 36 = 144

or, x  = 72°

⇒ ∠MLP = ∠MPL  = 72°

Now,as  JK  II  ML

⇒ ∠LJK = ∠JLM = 22° ( Alternate pair of angles)

Now, by the measure of straight angle:

∠MLP + ∠JLM + ∠JLK = 180°  ( Straight angle)

⇒ 72° + 22° + ∠JLK = 180°

or, ∠JLK  =  86°

In , in  ΔJLK

By ANGLE SUM PROPERTY:   ∠JKL + ∠JLK  + ∠LJK = 180°

⇒  ∠JKL + 86° + 22° = 180°

⇒ ∠JKL   = 180 - 108 = 72 , or ∠JKL = 72°

Hence, from  above proof ,  ∠MLP = 72° , ∠LJK = 22° , ∠JKL = 72° ,

∠KLJ  =  86°