Lesson 3: Use Properties of Angles, Triangles, and the Pythagorean Theorem

New Concept

This lesson introduces fundamental geometric rules. You'll learn how to use properties of angles, triangles, and the Pythagorean theorem ($a^2 + b^2 = c^2$) to find unknown lengths and angles, setting the stage for solving geometry problems.

What’s next

Now, let's apply these rules. You'll tackle interactive examples and practice problems to master finding missing angles and side lengths.

Property

If the sum of the measures of two angles is $180^\circ$, then the angles are supplementary.

If $\angle A$ and $\angle B$ are supplementary, then $m\angle A + m\angle B = 180^\circ$.

If the sum of the measures of two angles is $90^\circ$, then the angles are complementary.

Property

For any $\triangle ABC$, the sum of the measures of the angles is $180^\circ$.

Examples

• The measures of two angles of a triangle are $60^\circ$ and $85^\circ$. The third angle, $x$, is found by solving $60^\circ + 85^\circ + x = 180^\circ$, which gives $145^\circ + x = 180^\circ$, so $x = 35^\circ$.

Property

If two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths are in the same ratio.

Examples

• $\triangle ABC$ is similar to $\triangle XYZ$. If side $AB=6$ and corresponding side $XY=2$, the ratio is $\frac{6}{2}=3$. If side $BC=9$, then corresponding side $YZ$ must be $\frac{9}{3}=3$.

• A man who is 6 feet tall casts a 4-foot shadow. A nearby tree casts a 20-foot shadow. Let $h$ be the tree's height. The ratio is $\frac{h}{6} = \frac{20}{4}$, so $4h = 120$, and $h=30$ feet.

Property

In any right triangle $\triangle ABC$,

where $c$ is the length of the hypotenuse (the side opposite the right angle) and $a$ and $b$ are the lengths of the legs.