# 9.3 Use properties of angles, triangles, and the pythagorean theorem  (Page 3/15)

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## Right triangles

Some triangles have special names. We will look first at the right triangle    . A right triangle has one $\text{90°}$ angle, which is often marked with the symbol shown in [link] .

If we know that a triangle is a right triangle, we know that one angle measures $\text{90°}$ so we only need the measure of one of the other angles in order to determine the measure of the third angle.

One angle of a right triangle measures $\text{28°}.$ What is the measure of the third angle?

## Solution

 Step 1. Read the problem. Draw the figure and label it with the given information. Step 2. Identify what you are looking for. Step 3. Name. Choose a variable to represent it. Step 4. Translate. Write the appropriate formula and substitute. Step 5. Solve the equation.   Step 6. Check:  Step 7. Answer the question. One angle of a right triangle measures $\text{56°}.$ What is the measure of the other angle?

34°

One angle of a right triangle measures $\text{45°}.$ What is the measure of the other angle?

45°

In the examples 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. So 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 $\text{20°}$ more than the measure of the smallest angle. Find the measures of all three angles.

## Solution

 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. Now draw the figure and label it with the given information.    Step 4. Translate. Write the appropriate formula and substitute into the formula.  Step 5. Solve the equation.       Step 6. Check:  Step 7. Answer the question. The measure of one angle of a right triangle is $\text{50°}$ more than the measure of the smallest angle. Find the measures of all three angles.

20°, 70°, 90°

The measure of one angle of a right triangle is $\text{30°}$ more than the measure of the smallest angle. Find the measures of all three angles.

30°, 60°, 90°

## Similar triangles

When we use a map to plan a trip, a sketch to build a bookcase, or a pattern to sew a dress, we are working with similar figures. In geometry, if two figures have exactly the same shape but different sizes, we say they are similar figures    . One is a scale model of the other. The corresponding sides of the two figures have the same ratio, and all their corresponding angles are have the same measures.

The two triangles in [link] are similar. Each side of $\text{Δ}ABC$ is four times the length of the corresponding side of $\text{Δ}XYZ$ and their corresponding angles have equal measures. Δ A B C and Δ X Y Z are similar triangles. Their corresponding sides have the same ratio and the corresponding angles have the same measure.

## Properties of similar triangles

If two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths are in the same ratio. The length of a side of a triangle may be referred to by its endpoints, two vertices of the triangle. For example, in $\text{Δ}ABC\text{:}$

$\phantom{\rule{2em}{0ex}}\begin{array}{c}\text{the length}\phantom{\rule{0.2em}{0ex}}a\phantom{\rule{0.2em}{0ex}}\text{can also be written}\phantom{\rule{0.2em}{0ex}}BC\hfill \\ \text{the length}\phantom{\rule{0.2em}{0ex}}b\phantom{\rule{0.2em}{0ex}}\text{can also be written}\phantom{\rule{0.2em}{0ex}}AC\hfill \\ \text{the length}\phantom{\rule{0.2em}{0ex}}c\phantom{\rule{0.2em}{0ex}}\text{can also be written}\phantom{\rule{0.2em}{0ex}}AB\hfill \end{array}$

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