Exercise 8 Page 137

Let's recall the formula for the sum of the interior angle measures S of a polygon with n sides. In this formula, we subtract 2 from the number of sides n and then multiply the difference by 180^(∘). S=( n- 2)* 180^(∘) Whenever we want to find the sum of the interior measures of a polygon, we need to know the number of sides of the polygon. Then we can substitute this number for n into the formula and simplify the right-hand side.

We are given three angle measures of a quadrilateral and we want to find the measure of its fourth angle. To do so, we need to find the sum of the interior angle measures of a quadrilateral. Let's use the formula we wrote in Part A.

S=(n-2)*180^(∘)

Substitute

n= 4

S=( 4-2)*180^(∘)

SubTerm

Subtract term

S=2*180^(∘)

Multiply

S=360^(∘)

Now, we know that our quadrilateral has angles measuring 100^(∘), 90^(∘), and 90^(∘). Let x^(∘) represent the measure of the fourth angle. Because the sum of the interior angle measures in each quadrilateral is 360^(∘), the sum of the above angle measures must be equal to 360^(∘). Let's use this information to write an equation. 100^(∘)+ 90^(∘)+ 90^(∘)+ x^(∘)= 360^(∘) Finally, we can solve the equation to find the value of x.

100^(∘)+90^(∘)+90^(∘)+x^(∘)=360^(∘)

AddTerms

Add terms

280^(∘)+x^(∘)=360^(∘)

SubEqn

LHS-280^(∘)=RHS-280^(∘)

x^(∘)=80^(∘)

The measure of fourth angle of our quadrilateral is 80^(∘).

Let's take a look at the given pentagon. We are given that the sum of the measures of the angles of the pentagon is 540^(∘).

Now, we want to divide this pentagon into triangles. To do so, we can draw two different segments from any of the vertices. Let's do it!

We divided our pentagon into 3 triangles. Recall that the sum of the interior angles of each triangle is 180^(∘). 3* 180^(∘)=540^(∘) Notice that the product of the number of triangles and the sum of the measures of the angles in a triangle is equal to 540^(∘). Now let's see what happens when we divide our pentagon into more than 3 triangles — say, for example, 4 triangles.

If we use the same logic as before, we multiply 180^(∘) by 4. 4* 180^(∘)=720^(∘) It seems like this time we did not get the sum of measures of a pentagon. However, look carefully at the figure again. Notice that three out of four triangles have one angle that is not a part of pentagon's interior angles.

This means that we should subtract the measures of these interior angles from our expression to get the sum of measures of our pentagon. Since the three angles form a line, together they form a straight angle. This means that the sum of their measures is 180^(∘). 4* 180^(∘)- 180^(∘)=540^(∘) Now we ended up with the correct answer! Remember that if the pentagon is divided into more than 3 triangles, the measures of the interior angles should be subtracted from the total measure of angles to find the sum of pentagon's angle measures.

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Exercise 8 Page 137

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