Answer:
x=27°
Step-by-step explanation:
we know that
The Triangle Exterior Angle Theorem, states that: An exterior angle of a triangle is equal to the sum of the opposite interior angles
step 1
Find the measure of angle PLM
we know that
m∠PLM+m∠LPM=m∠PMN ----> by Triangle Exterior Angle Theorem
we have
m∠LPM=x°
m∠PMN=2x+72°
substitute
m∠PLM+x=2x+72°
m∠PLM=2x-x+72°
m∠PLM=x+72°
step 2
Find the measure of angle x
we know that
3x+m∠PLM=180° ----> by supplementary angles (form a linear pair)
we have
m∠PLM=x+72°  (see step 1)
substitute
3x+x+72°=180°
4x=180°-72°
4x=108°
x=27°
Answer:
By the given diagram,
PLM is a triangle,
K and N are points on line MN,
m∠KLP = 3x, m∠P = x, m∠PMN = 2x + 72°
We need to give the reasons in the steps of finding the value of x.
For this we have to know the following properties :
Linear pairs: Adjacent angles which are supplementary.
Subtraction property of equality : We can subtract a number or expression in both sides of an equation.
i.e. a = b ⇒ a + c = b + c
Exterior angle theorem : exterior angle formed by extending the side of a triangle is equal to the sum of its non-adjacent angles.
Thus, the steps of finding value of x are as follow,
1. m∠KLP+m∠PLM = 180°    ( linear pairs )
2. 3x +m∠PLM = 180°        ( Substitution)
3. m∠PLM = 180° - 3x        ( Subtraction property of equality)
4. m∠PMN=m∠P+m∠PLM    ( Exterior angle theorem )
5. 2x + 72° = x + 180° - 3x     ( Substitution)
6. 2x + 72° = 180° - 2x        ( Solving )
  2x = 180° - 2x - 72° Â
  2x + 2x = 108°
  4x = 108°
  x  = 27°
