What is the measure of the vertex angle of an isosceles triangle if the measure of the base angle is 36 degrees?

There are many types of triangles in the world of geometry. There is a special triangle called an isosceles triangle. In an isosceles triangle, the base angles have the same degree measure and are, as a result, equal (congruent). Similarly, if two angles of a triangle have equal measure, then the sides opposite those angles are the same length. The easiest way to define an isosceles triangle is that it has two equal sides.

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In an isosceles triangle, we have two sides called the legs and a third side called the base. The angle located opposite the base is called the vertex.

Sample A:

The vertex angle B of isosceles triangle ABC is 120 degrees. Find the degree measure of each base angle.

Solution:

(1) Let x = the measure of each base angle. (2) Set up an equation and solve for x. base angle + base angle + 120 degrees = 180 degrees x + x + 120 degrees = 180 degrees 2x + 120 = 180 2x = 180 - 120 2x = 60 x = 60/2

x = 30

Each base angle of triangle ABC measures 30 degrees.

Sample B:

In isosceles triangle RST, angle S is the vertex angle. Base angles R and T both measure 64 degrees. Find the degree measure of the vertex angle S.

Solution:

(1) Let x = measure of vertex angle S.
(2) Set up an equation and solve for x.

base angle + base angle + vertex angle S = 180 degrees 64 degrees + 64 degrees + x = 180 degrees 128 + x = 180 x = 180 - 128

x = 52

The measure of vertex angle S in triangle RST is 52 degrees.

Sample C:

The degree measure of a base angle of isosceles triangle XYZ exceeds three times the degrees measure of the vertex Y by 60. Find the degree measure of the vertex angle Y. Notice that it's hard to draw a picture without knowing which angles are largest.

We need to make an equation out of this problem, so let's figure out what it's trying to tell us. First we read "The degree measure of a base angle", so let's start with X=

Our equation so far: X=

Now we see "exceeds three times... Y... by 60", which means 3Y + 60.

Our equation now: X = 3Y + 60

Since we know that X = Z because it is an isosceles triangle, then we can solve for the measures of all the angles.

base angle + base angle + vertex = 180 X + Y + Z = 180 (3Y + 60) + Y+ (3Y + 60) = 180 7Y + 120 = 180 7Y = 60 Y = 60/7

Y = 8.57 degrees

The vertex angle Y of triangle XYZ equals 8.57 degrees.

Lesson provided by Mr. Feliz

Isosceles triangle calculator is the best choice if you are looking for a quick solution to your geometry problems. Find out the isosceles triangle area, its perimeter, inradius, circumradius, heights and angles - all in one place. If you want to build a kennel, find out the area of Greek temple isosceles pediment or simply do your maths homework, this tool is here for you. Experiment with the calculator or keep reading to find out more about the isosceles triangle formulas.

An isosceles triangle is a triangle with two sides of equal length, which are called legs. The third side of the triangle is called base. Vertex angle is the angle between the legs and the angles with the base as one of their sides are called the base angles.

Here are the most important properties of isosceles triangles:

It has an axis of symmetry along its vertex height;
Two angles opposite the legs are equal in length; and
The isosceles triangle can be acute, right, or obtuse, but it depends only on the vertex angle (base angles are always acute)

The equilateral triangle is a special case of an isosceles triangle. You can learn about all the possible types of triangles in the classifying triangles calculator.

To calculate the isosceles triangle area, you can use many different formulas. The most popular ones are the equations:

Given leg a and base b:

area = (1/4) × b × √( 4 × a² - b² )
Given h height from apex and base b or h2 height from other two vertices and leg a:

area = 0.5 × h × b = 0.5 × h2 × a
Given any angle and leg or base

area = (1/2) × a × b × sin(base_angle) = (1/2) × a² × sin(vertex_angle)

Also, you can check our triangle area calculator to find out other equations, which work for every type of triangle, not only for the isosceles one.

To calculate the isosceles triangle perimeter, simply add all the triangle sides:
perimeter = a + a + b = 2 × a + b

Isosceles triangle theorem, also known as the base angles theorem, claims that if two sides of a triangle are congruent, then the angles opposite to these sides are congruent.

Also, the converse theorem exists, stating that if two angles of a triangle are congruent, then the sides opposite those angles are congruent.

A golden triangle, which is also called sublime triangle is an isosceles triangle in which the leg is in the golden ratio to the base:

a / b = φ ~ 1.618

The golden triangle has some unusual properties:

It's the only triangle with three angles in 2:2:1 proportions
It's the shape of the triangles found in the points of pentagrams
It's used to form a logarithmic spiral

Let's find out how to use this tool on a simple example. Have a look at this step-by-step solution:

Determine what is your first given value. Assume we want to check the properties of the golden triangle. Type 1.681 inches into leg box.
Enter second known parameter. For example, take a base equal to 1 in.
All the other parameters are calculated in the blink of an eye! We checked for instance that isosceles triangle perimeter is 4.236 in and that the angles in the golden triangle are equal to 72° and 36° - the ratio is equal to 2:2:1, indeed.

You can use this calculator to determine different parameters than in the example, but remember that there are in general two distinct isosceles triangles with given area and other parameter, e.g. leg length. Our calculator will show one possible solution.

To compute the area of an isosceles triangle with leg a and base b, follow these steps:

Apply the Pythagorean theorem to find the height: √( a² - b²/4 ).
Apply the standard triangle area formula, i.e., multiply base b by the height found in Step 1 and then divide by 2.
That is, the final formula we got is:

area = ½ × b × √( a² - b²/4 )

We compute the perimeter of an isosceles triangle with leg a and base b with the help of the formula perimeter = 2 × a + b. This formula makes use of the fact that the two legs of an isosceles triangle are of equal length.

The answer is 6.93. To derive it, we can use the formula area = ½ × b × √( a² - b²/4 ) with a = b = 4.

Alternatively, we can notice that, in fact, we have here an equilateral triangle: the are formula simplifies to area = a² × √3 / 4 with a = 4.

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