Question

Find the value of x in each case. Give reasons to justify your solutions!

Answer 1

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°  

   2x + 2x = 108°

   4x = 108°

   x   = 27°

Answer 2

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°