Exercise 15 Page 420

We want to find the value of x in the given picture.

We can see that we do not know the measure of the third angle in either of the two triangles formed by the roads. However, if either of the triangles is a right triangle, then we will know that its missing angle is a right angle. To check if either of these is a right triangle, we will use the Converse of the Pythagorean Theorem.

Converse of the Pythagorean Theorem

For a triangle with sides of lengths a, b, and c, if the equation a^2+b^2=c^2 is true, then the triangle is a right triangle.

Now, we will highlight one of the triangles created by the five roads and then check if it is a right triangle. We will label the vertices of the triangle as A, B, and C. Let's take a look at the graph using the assigned labels.

From the graph, we can see that the lengths of the sides in triangle ABC are 73.2, 97.6, and 122 feet. We will substitute these values into the equation a^2+b^2=c^2 and verify if it holds true. Keep in mind that we always substitute the longest side for c. We found that the equation is true for sides of the triangle ABC. This means that the triangle ABC is a right triangle by the Converse of the Pythagorean Theorem.

Next, let's recall that measures of the interior angles in a triangle sum up to 180^(∘). We can use this information to find the value of x. Remember that the measure of a right angle is 90^(∘). x^(∘)+37^(∘)+ 90^(∘)=180^(∘) ⇕ x^(∘)=53^(∘) The value of x is 53.