Andre and Clare are discussing the triangle \(ABC\) that has a right angle at \(C\) and a hypotenuse of length 15 units. Andre thinks the triangle could have legs that are 9 and 12 units long. Clare thinks angle \(B\) could be 20 degrees and then side \(BC\) would be 14.1 units long. Do you agree with either of them? Explain or show your reasoning.

A triangle has sides with lengths 5, 12, and 13.

1. Verify this is a right triangle.
2. Approximate the acute angles in this triangle.

Approximate the angles that have the following quotients: 

1. adjacent leg \(\div\) hypotenuse = 0.966
2. opposite leg \(\div\) hypotenuse = 0.469
3. adjacent leg \(\div\) hypotenuse = 0.309
4. opposite leg \(\div\) adjacent leg = 1.036

Lin missed class, and Tyler is helping her use the table to approximate the angle measures that have the ratios listed. Tyler says, “You can use the Right Triangle Table to figure this out.” Lin notices that some of the ratios are the same in each row. Estimate the angles, and explain why some of the values are repeated.

Elena is proving the Pythagorean Theorem. She knows the goal is to prove that \(a^2 + b^2 = c^2\) in a right triangle. So far she has this:

In a right triangle the altitude that intersects the hypotenuse decomposes the triangle into two smaller right triangles. These triangles are similar to the large triangle by the Angle-Angle Triangle Similarity Theorem since each smaller triangle shares one angle with the larger triangle and has a right angle. Similar triangles have proportional side lengths, so \(\underline{\hspace{.5in}1\hspace{.5in}}\) and \(\underline{\hspace{.5in}2\hspace{.5in}}\). I can rewrite those equations to get \(a^2=xc\) and \(b^2=yc\). Therefore \(a^2+b^2=\) \(\underline{\hspace{.5in}3\hspace{.5in}}\).

Fill in the blanks, and finish the proof Elena started.