A right triangle is a triangle that has one right angle. The side opposite the right angle is always the longest and is known as the hypotenuse. The other sides are commonly called legs. Notice that in a right triangle, the legs are perpendicular to each other. If one of the acute angles in the triangle is labeled, the legs can be discussed relative to that angle. Consider an acute angle labeled as $\angle \theta$ on the diagram. The side that forms the angle is the adjacent side and the side not touching the angle is the opposite side. Note that the opposite and adjacent sides change when $\angle \theta$ changes, but the hypotenuse is always the same.

A Pythagorean triple, commonly written as $(a,b,c),$ is a set of three natural numbers that satisfy the Pythagorean Theorem. A right triangle can be drawn using the numbers of a Pythagorean triple as its side lengths. The lowest valued set of numbers that form a Pythagorean triple are $3,$ $4,$ and $5.$ There are infinitely many Pythagorean triples. The following table shows a few.

Triple	Substitute in $a^2+b^2=c^2$	Simplify
$(3,4,5)$	$3^2+4^2\stackrel{?}{=}{5}^2$	$9+16=25✓$
$(5,12,13)$	$5^2+{12}^2\stackrel{?}{=}{13}^2$	$25+144=169✓$
$(8,15,17)$	$8^2+{15}^2\stackrel{?}{=}{17}^2$	$64+225=289✓$
$(7,24,25)$	$7^2+{24}^2\stackrel{?}{=}{25}^2$	$49+576=625✓$

If $(a,b,c)$ is a Pythagorean triple, then so is $(ka,kb,kc)$ for any natural number $k.$

$(a,b,c)$	$k$	$(ka,kb,kc)$
$(3,4,5)$	$3$	$(3(3),3(4),3(5))$ $\Updownarrow$ $(9,12,15)$
$(5,12,13)$	$2$	$(2(5),2(12),2(13))$ $\Updownarrow$ $(10,24,26)$

A $30\text{-}60\text{-}90$ triangle is a special type of right triangle. It has two acute angles measuring $30^{\circ }$ and $60^{\circ }$ in addition to the right angle.

If the shorter leg in a $30\text{-}60\text{-}90$ triangle has the length $\ell ,$ the longer leg will have the length $\sqrt{3}\ell ,$ and the length of the hypotenuse will be $2\ell .$

These relations can be deducted from the trigonometric ratios and the known trigonometric values of the notable angles.

A $45\text{-}45\text{-}90$ triangle is a special type of right triangle. It has two acute angles measuring $45^{\circ }$ and one right angle.

Since these two acute angles are congruent, its legs are also congruent. This implies that the triangle is isosceles. If the legs have the length $\ell ,$ the length of the hypotenuse will be $\sqrt{2}\ell .$

This relation can be deducted from the trigonometric ratios and the known trigonometric values of the notable angles.