## Table of Contents

- What is a Right Triangle?
- Right Triangle Properties
- Right Triangle Proof
- Right Triangle Theorems
- Lesson Summary

What is a right triangle? Learn its definition and properties. See the Pythagorean Theorem and the Right Triangle Altitude Theorem, and use them in proofs.
Updated: 09/29/2021

- What is a Right Triangle?
- Right Triangle Properties
- Right Triangle Proof
- Right Triangle Theorems
- Lesson Summary

The definition of a **right triangle** is a triangle in which one of the interior angles measures 90 degrees. This 90-degree interior angle is also referred to as a **right angle**.

Triangles can either be equilateral, isosceles, or scalene. All triangles, regardless of type or size, have three interior angles totaling 180 degrees.

**Scalene Triangle**: a triangle whose sides are all different lengths. Its interior angles are all different.**Isosceles Triangle**: a triangle with two equal sides and two equal angles.**Equilateral Triangle**: all three sides of this triangle are equal, and all its interior angles are 60 degrees.

Since a right triangle must have a 90-degree angle, and all interior angles of an equilateral triangle are 60 degrees, right triangles cannot be equilateral. Right triangles must be either isosceles or scalene.

One of the interior angles of an *isosceles right triangle* will be 90 degrees. Since the other two interior angles must sum to 90, and they must be equal given the properties of isosceles triangles, the other two will be 45-degree angles, as seen in the illustration below.

The second type of right triangle is *scalene*. One of its angles will be 90 degrees, and the other two will sum to 90 degrees. All three sides of these triangles will be different lengths, and all three angles will be different.

30-60-90 triangles fall under this category. As its name implies, a 30-60-90 triangle is one in which the three interior angles are 30, 60, and 90 degrees.

30-60-90 triangles are considered **special triangles** because we find interesting qualities when examining these triangles.

The lengths of the side of a 30-60-90 triangle are consistent and can be described with the following ratio:

{eq}1 : 2 : \sqrt{3} {/eq}

If the length of the short side (opposite the 30-degree angle) can be represented by the variable x, the hypotenuse is of length 2x, meaning that it is twice as long as the short side, and the long leg is of length {eq}\sqrt{3} {/eq}

**Right triangles** have many significant properties.

- Like all triangles, the sum of the three interior angles is 180 degrees.
- The widest interior angle is 90 degrees, also known as a
*right angle*. - The sum of the two smaller angles will be 90 degrees, making them
**complementary angles**. This makes it possible for one to be solved using the value of the other. - In an
*isosceles*right triangle, the two complementary angles will each be 45 degrees.

- The side opposite the 90-degree angle, known as the
**hypotenuse,**will be the longest side. - The other two sides, known as the
**legs**, will be smaller proportionate to their opposite sides.

- If one multiplies the lengths of the legs of a right triangle and divides the product by 2, we obtain the Area of the right triangle. {eq}A = \frac{1}{2}\ bh {/eq} where b is the base and h is the height of the triangle.

If we add together the squares of the two legs of a right triangle, we obtain the square of the *hypotenuse*. (Pythagorean Theorem.)

- Right Triangle Trigonometry begins with an analytical look at the right triangle, as would be expected. Consider the right triangle in the illustration below. One acute angle is placed at the origin of an XY axis, and the right angle lies elsewhere on the X-axis. The acute angle at the origin of the XY axis is labeled angle ? (theta), and the three sides are labeled Adjacent, Opposite, and Hypotenuse. Notice that the adjacent side lies on the x-axis and that the opposite side is opposite to angle ?. We are now able to establish relationships between the sides and angles of this right triangle using
**Trigonometric Ratios:**

Function Name................Abbreviation................Value

Sine of theta................{eq}sin \theta {/eq} ................Opposite / Hypotenuse

Tangent of theta................{eq}tan \theta {/eq} ................Opposite / Adjacent

Secant of theta................{eq}sec \theta {/eq} ................Hypotenuse / Adjacent

Cosine of theta................{eq}cos \theta {/eq} ................Adjacent / Hypotenuse

Cosecant of theta................{eq}csc \theta {/eq} ................Hypotenuse / Opposite

Cotangent of theta................{eq}cot \theta {/eq} ................Adjacent / Opposite

By the Pythagorean theorem, the square of the hypotenuse of a right triangle equals the sum of the squares of the two legs. Expressed as an equation,

{eq}c^{2}\ = a^{2} + b^{2} {/eq}

If the Pythagorean theorem is true, then the converse must also be true. That is, if the square of the longest side of a triangle equals the sum of the squares of the other two sides, it must be a right triangle.

Suppose a triangle whose apparent hypotenuse, or the longest side, measures 12 units in length, and the two adjacent sides are 8 and 9 units. If we square 8, we obtain 64, and if we square 9, we obtain 81. Adding 64 and 81, we obtain 145. When we square 12, we obtain 144, which clearly does not equal 145. Therefore, what appears to be a right angle, in fact, is not.

Suppose, however, that a right triangle has a hypotenuse measuring 17 units in length, with two adjacent legs measuring 8 units and 15 units. Squaring 17, one obtains 289. Squaring 8 and 15, one obtains 64 and 225, respectively. The result of adding 64 and 225 is 289, which is exactly the square of 17. Therefore, a triangle with sides of 17, 8, and 15 units of length must be a right triangle.

If it can be shown that the two legs of a right triangle are congruent to the two corresponding legs of another right triangle, we know that the two triangles are congruent, as seen in the illustration below. This is *leg-leg congruence*.

If it can be shown that the hypotenuse and acute angle of a right triangle are congruent, we know that the two triangles are congruent, as seen in the illustration below. This is *hypotenuse-angle congruence.*

If it can be shown that one leg and an acute angle of a right triangle are congruent to the corresponding leg and the acute triangle of another right triangle, we know the two right triangles are congruent, as seen in the illustration below. This is *leg-angle congruence.*

If it can be shown that the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and the corresponding leg of another right triangle, we know the two right triangles are congruent, as seen in the illustration below. This is *hypotenuse-leg congruence.*

The Pythagorean Theorem states that the square of the hypotenuse of a right triangle equals the sum of the squares of the two legs. Another way to put this is:

{eq}c^{2} = a^{2} + b^{2} {/eq}

where c is the hypotenuse and a and b are the two legs.

The **Pythagorean Theorem** is extremely useful in Trigonometry, but it is also very useful in building construction. For example, when a house or building is being built, the construction workers can measure two adjacent sides of the building, then a diagonal from one side to its adjacent side. If the square of the diagonal is equal to the sum of the squares of the two adjacent sides, the construction workers know that that the building is *square*, or at a perfect right angle.

Suppose a right triangle was to be laid out so that it lies flat on its hypotenuse, as shown below:

A line CD drawn from the apex of this right triangle to a point perpendicular to the hypotenuse creates two right triangles similar to the original right triangle. See the Illustration below.

This perpendicular then divides the mother right triangle ABC into two daughter triangles, ACD and BCD. It is interesting to note that the original, or mother triangle had interior angular measures of 90, 30, and 60 and that the two daughter triangles have the same interior angular measures. One says, then, that the two daughter triangles are similar to the mother triangle, because the interior angles are congruent, although the side lengths are not, as seen in the image below.

A **right triangle** is one that has a right angle.

Right triangles may be **isosceles** or **scalene**. Right isosceles triangles have two sides of equal length, while right scalene triangles have none.

Right triangles have properties that distinguish them from other triangles. In particular, the Pythagorean Theorem only applies to right triangles.

**30-60-90 triangles** are **special triangles** due to certain qualities they exhibit, most particularly the ratio of the sides of such triangles.

**Trigonometric Ratios** can be derived from a right triangle. Trigonometric ratios allow us to solve right triangles in unique ways, such as determining the length of one of the sides of the triangle, given the measure of the opposite angle and the length of an adjacent side.

The **Pythagorean Theorem** states that the square of a hypotenuse of a right triangle is equal to the sum of the squares of the two adjacent sides.

A triangle may be proven to be a right triangle by employing the converse of the Pythagorean theorem: that is, if the square of the hypotenuse is equal to the sum of the squares of the two adjacent sides, we know the triangle is a right triangle.

Two right triangles may be proven to be congruent by employing leg-leg, hypotenuse-angle, leg-angle, and hypotenuse-leg congruence principles.

A right triangle may be divided into two similar right triangles by drawing a line from the point at which the right angle is found to a point perpendicular to the hypotenuse of the right triangle.

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Frequently Asked Questions

The missing side of the triangle is calculated using the Pythagorean theorem that is (Hypotenuse)^2=(Base)^2+(Altitude) ^2. If two sides from three sides of the right-angle triangle are known, then the other sides can be easily found by using this formula.

One angle of a right triangle always measures to be 90 degrees.

The hypotenuse is the longest side of the right-angle triangle.

The side that is opposite to the 90-degree angle is the hypotenuse.

The sum of the other two interior angles of a right-angle triangle is always 90 degrees.

The side adjacent to the 90 degrees or right angle in the triangle is known as the base and altitude of the triangle.

There are three sides in a right triangle; the base and altitude are the sides nearest the 90-degree angle, and opposite of the 90-degree angle is the hypotenuse. In trigonometrical words, the base and altitude are called adjacent and opposite sides. The term hypotenuse is the same everywhere.

A right triangle is a triangle where one of the interior angles is 90 degrees; any one of the angles must be a right angle and the sum of the three angles must be equal to 180 degrees. So, it is also named as 90-degree angle triangle.

Any square-shaped object that is cut diagonally will result in two right-angle triangles. If horizontal and vertical lines are drawn through a kite or any object that is in a square or rhombus shape, the angle at the intersection is 90 degrees and will have four right-angle triangles in it.

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