Exercise 41 Page 432

In order to list the side length of the triangles from shortest to longest, we start with analyzing the given diagram.

We are given the measures of all the angles except two. Let's find them! We can use the fact that the sum of triangle angles is 180^(∘). Applied to △ ABC, it can be written the following way. m∠ A+ m∠ ABC+ m∠ ACB=180^(∘) It is given that ∠ A measures 62^(∘) and ∠ ABC measures 71^(∘). Let's substitute these values into the above equality and calculate m∠ ACB.

Now, we can apply this fact to triangle △ BCD to find m∠ CBD. m∠ CBD+ m∠ BCD+ m∠ D=180^(∘) If we substitute m∠ BCD with 83^(∘) and m∠ D with 42^(∘), we will get an equation with the only unknown m∠ CBD.

Now that we know the measures of all of the angles, we can use Theorem 5.10, which states the following. If one angle of a triangle has a greater measure than another angle, then the side opposite to the greater angle is longer than the side opposite to the lesser angle. According to this theorem, we need first to order the angles from least to greatest. Let's do this!

Comparing the measures of the angles, we can list them the following way. ∠ D, ∠ ACB, ∠ CBD, ∠ A, ∠ ABC, ∠ BCD Now, using the diagram, we can find the opposite sides to these angles. ∠ D - BC ∠ ACB - AB ∠ CBD - CD ∠ A - BC ∠ ABC - AC ∠ BCD - BD By the means of the above theorem, the order of the sides from shortest to longest is the same as the order of the angles opposite to them. BC, AB, CD, BC, AC, BD