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One angle of a right triangle measures $56\text{\xb0}.$ What is the measure of the other small angle?
$34\text{\xb0}$
One angle of a right triangle measures $45\text{\xb0}.$ What is the measure of the other small angle?
$45\text{\xb0}$
In the examples we have seen so far, we could draw a figure and label it directly after reading the problem. In the next example, we will have to define one angle in terms of another. We will wait to draw the figure until we write expressions for all the angles we are looking for.
The measure of one angle of a right triangle is 20 degrees more than the measure of the smallest angle. Find the measures of all three angles.
Step 1. Read the problem.  
Step 2. Identify what you are looking for.  the measures of all three angles 
Step 3. Name. Choose a variable to represent it. 
$\phantom{\rule{0.24em}{0ex}}$ Let
$a={1}^{\text{st}}$ angle.
$a+20={2}^{\text{nd}}$ angle $\phantom{\rule{1.7em}{0ex}}90={3}^{\text{rd}}$ angle (the right angle) 
Draw the figure and label it with the given information  
Step 4. Translate  
Write the appropriate formula.
Substitute into the formula. 

Step 5. Solve the equation. 
55 90 third angle 
Step 6. Check.
$\begin{array}{ccc}\hfill 35+55+90& \stackrel{?}{=}\hfill & 180\hfill \\ \hfill 180& =\hfill & 180\u2713\hfill \end{array}$ 

Step 7. Answer the question.  The three angles measure 35 ^{°} , 55 ^{°} , and 90 ^{°} . 
The measure of one angle of a right triangle is 50° more than the measure of the smallest angle. Find the measures of all three angles.
$20\text{\xb0},70\text{\xb0},90\text{\xb0}$
The measure of one angle of a right triangle is 30° more than the measure of the smallest angle. Find the measures of all three angles.
$30\text{\xb0},60\text{\xb0},90\text{\xb0}$
We have learned how the measures of the angles of a triangle relate to each other. Now, we will learn how the lengths of the sides relate to each other. An important property that describes the relationship among the lengths of the three sides of a right triangle is called the Pythagorean Theorem . This theorem has been used around the world since ancient times. It is named after the Greek philosopher and mathematician, Pythagoras, who lived around 500 BC.
Before we state the Pythagorean Theorem, we need to introduce some terms for the sides of a triangle. Remember that a right triangle has a $90\text{\xb0}$ angle, marked with a small square in the corner. The side of the triangle opposite the $90\text{\xb0}$ angle is called the hypotenuse and each of the other sides are called legs .
The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the lengths of the two legs equals the square of the length of the hypotenuse. In symbols we say: in any right triangle, ${a}^{2}+{b}^{2}={c}^{2},$ where $a\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b$ are the lengths of the legs and $c$ is the length of the hypotenuse.
Writing the formula in every exercise and saying it aloud as you write it, may help you remember the Pythagorean Theorem.
In any right triangle, ${a}^{2}+{b}^{2}={c}^{2}.$
where a and b are the lengths of the legs, c is the length of the hypotenuse.
To solve exercises that use the Pythagorean Theorem, we will need to find square roots. We have used the notation $\sqrt{m}$ and the definition:
If $m={n}^{2},$ then $\sqrt{m}=n,$ for $n\ge 0.$
For example, we found that $\sqrt{25}$ is 5 because $25={5}^{2}.$
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