Exercise 23 Page 330

In order to find the measure of $∠ P N M,$ let's analyze the given diagram.

As we can see,

∠ P N M

is the sum of the angles

∠ P N Q

and

∠ Q N M .

By the Angle Addition Postulate, its measure is the sum of these angles measures.

m ∠ P N M = m ∠ P N Q + m ∠ Q N M

Let's substitute

m ∠ P N Q

with

(4 x - 8)^{\circ}

and

m ∠ Q N M

with

(3 x + 5)^{\circ},

and find the expression that represents

m ∠ P N M .

m ∠ P N M = m ∠ P N Q + m ∠ Q N M

SubstituteII

$m ∠ P N Q = 4 x - 8$ , $m ∠ Q N M = 3 x + 5$

m ∠ P N M = 4 x - 8 + 3 x + 5

AddSubTerms

Add and subtract terms

m ∠ P N M = 7 x - 3

To evaluate the measure of the angle, we need to know

x .

Let's try to find it.

From the diagram, we can see $\overline{M Q} ⊥ N M$ and $\overline{Q P} ⊥ N P .$ Thus, $Q M$ and $Q P$ are the distances from the point $Q$ to the sides of the angle $∠ P N M .$ Also, both segments measures $18,$ so $Q$ is equidistant from $N M$ and $N P .$ Now we can use the Converse of the Angle Bisector Theorem.

Converse of the Angle Bisector Theorem
If a point in the interior of an angle is equidistant from the sides of the angle, then it is on bisector of the angle.

According to this theorem,

N Q,

on which

Q

lies, is a bisector of

∠ P N M .

This means that angles

∠ P N Q

and

∠ Q N M

are congruent. Hence, we can set their measures equal.

4 x - 8 = 3 x + 5

This way we get an equation where the only unknown is

x .

Let's solve!

4 x - 8 = 3 x + 5

SubEqn

$LHS - 3 x = RHS - 3 x$

x - 8 = 5

AddEqn

$LHS + 8 = RHS + 8$

x = 13

Now that we found the value of

x,

we can finally calculate the measure of

∠ P N M .

Let's substitute its value into the expression for

m ∠ P N M

that we found earlier.

m ∠ P N M = 7 x - 3

Substitute

$x = 13$

m ∠ P N M = 7 (13) - 3

Multiply

m ∠ P N M = 91 - 3

SubTerm

Subtract term

m ∠ P N M = 88

Therefore,

∠ P N M

measures

8 8^{\circ} .

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