# Acute Angles

Acute angles are angles that measure less than 90o.  From the Latin term, acutus, which means something sharp or pointed, acute angles are known for their pointed shape. More often than not, the acute angles that you’ll encounter exhibit a V-shape. Try to think of real-world examples that exhibit this shape and you’ve probably already encountered acute angles before.

In this article, we’ll show you the components and basic definition of acute angles. We’ll also show you angle measures and real-world examples of acute angles so that you’re more familiar with how acute angles look. This discussion will also cover acute triangles’ definition and properties – showing you how important acute angles are in the foundation of geometry.

## What Are Acute Angles?

Acute angles are angles that measure less than 90o or simply, the right angles. This means that any angle measuring between 0o and 90o is considered an acute angle. Let’s take a look at the acute angle shown below.

We can see that an acute angle is formed by two arms or rays and they meet at a common point we call the vertex. For the diagram shown above, the acute angle, \angle AOB measuring \theta degrees, is formed by the rays, \overline{AB} and \overline{AO}. For \angle AOB to be an acute angle, \theta must be between 0o and 90o.

To better understand what makes acute angles unique, let us show you the three common types of angles: acute, right, and obtuse angle.

By showing you how an acute angle looks like compared to the right and obtuse angles, it helps you understand and identify acute angles from one look.

• We can see that acute angles lean in towards the right direction.
• Right angles will always form an “L”-shape.
• Obtuse angles go beyond the “L”-shape and lean towards the left direction.

You’ve probably also encountered acute angles in the past without knowing that they contain acute angles. The images shown above are three of the many examples of acute angles in the real world.

• A pizza sliced into 6 or 8 slices will have slices that contain acute angles. This makes sense because each slice will have angle measures of either 45o or 60o, respectively.
• Triangular signs like the one shown above will contain three interior angles that are each acute angle.
• A pair of opened scissors can also form acute angles when not opened too widely.

You can still think of a few more examples on your own. It’s important that you get used to identifying the shape formed by acute angles. For now, let’s try out a warm-up problem to test your understanding of acute angles.

### Problem 1

In the images shown below, you’ll see a protractor – a handy tool we can use to measure and construct angles. For each item, identify whether the angle highlighted is indeed acute or not.

a.

The angle’s arm lands on 30o, so the measure of \angle AOB is 30o. Recall that all acute angles measure between 0 and 90o, so our angle is an acute angle. In addition to its measure, the shape of the angle also confirms that it’s an acute angle.

b.

By inspection, it might appear tricky to identify this angle since it is close to being a right angle. But, by seeing its actual measure, we’ll confirm whether the angle is acute or not. Since the final position of \angle AOB’s arm lands on 85o. Since 85\degree < 90\degree, \angle AOB is an acute angle.

c.

From its sketch alone, we can see that \angle AOB is leaning towards the left side and past the 90o mark. The protractor measures \angle AOB as 120o. This is beyond the angles that can be defined as an acute angle, so \angle AOB is not an acute angle. In fact, it’s an obtuse angle.

Now that we’ve covered the fundamentals of what makes acute angles unique, let’s extend our knowledge by exploring acute angles’ properties and applications.

## Other Properties of Acute Angles

Acute angles exhibit interesting properties that will come in handy later on.  Let’s cover these bases for you to make sure you know all the fundamentals of acute angles.

• Two acute angles can be complementary (add up to 90o) but can never be supplementary (add up to 180o).
• This also means that two acute angles can form a right triangle but never a straight angle.
• When a polygon has five or more vertices or sides, the exterior angle of the polygon is an acute angle.
• When all the interior angles of a triangle are acute, we call it an acute triangle.

### Problem 2

Determine whether the following statements are true or false.

a. A hexagon will always have exterior angles that are each acute angle.

We’ve mentioned that polygons that contain five or more sides such as hexagons (six-sided polygons) will always have acute angles for their exterior angles. For example, if we have the regular hexagon shown below, we have the exterior angles shown below.

This confirms that the statement is true.

b. When a right angle is divided by an angle bisector, it will return one acute angle and one obtuse angle.

We know that a right angle will always have an angle measure of 90o. When an angle bisector divides an angle, it divides it into two equal angles.

This means that we’ll divide 90o by 2, so each angle will have a measure of 45o. This is less than 90o (this is expected since we’re dividing 90o by 2), so we can confirm that the resulting angles will be acute making the statement true.

Now that we know some interesting facts about acute angles, let’s now zone in on acute angle triangles. These triangles exhibit interesting behaviors as well due to the fact that they contain acute angles.

## What Are Acute Angle Triangles?

As we have pointed out, acute triangles are simply triangles that contain three acute angles within. This means that an acute triangle will have interior angles each measuring less than 90o. Here are some examples of acute triangles:

Each acute triangle contains acute angles within its interior angles. Take a look at the third triangle: we call this an equilateral triangle. It’s a special type of acute triangle that has interior angles each measuring 60o.

This means that when a triangle contains three acute angles, we can immediately identify it as an acute triangle. This triangle also exhibits interesting properties in terms of its sides – we call this relationship the acute angle triangle formula.

## Acute Angle Triangle Formula

The sides of any acute angle triangle satisfy a particular relationship defined by the acute triangle formula. The acute angle triangle formula is similar to Pythagorean’s theorem (a^2 + b^2 = c^2).

Suppose that we have the acute triangle, \Delta ABC, as shown above, the sides of these triangles will satisfy the following inequalities.

\begin{aligned}a^2 + b^2 &> c^2\\ b^2 + c^2 &> a^2\\ c^2 + a^2 &> b^2  \end{aligned}

This is why we can also call this formula the triangle inequality for acute triangles. What this acute triangle formula means in words is that the square of the shorter sides’ sum will always be larger than the square of the longest sign. Let’s now apply what we’ve learned and worked on the problems below.

### Problem 3

Show that it is possible for an acute angle to have the following sides as shown below.

Recall that an acute triangle satisfies the inequality shown below.

\begin{aligned}a^2 + b^2 &> c^2\end{aligned}

For this inequality, we have a = 10 ft, b = 8 ft, and c = 12 ft. Let’s check if the sides satisfy the inequality by substituting the values into the inequality.

\begin{aligned}a^2 + b^2 &> c^2\\10^2 + 8^2 &> 12^2\\100 + 64 &> 144\\164 &> 144\\&\Rightarrow \text{Acute Triangle} \end{aligned}

Since the sides satisfy the inequality, the sides are valid and can form an acute triangle.

### Problem 4

Another special acute triangle is the golden triangle – it’s an acute isosceles triangle. We call it the golden triangle because it’s the only triangle that contains angles with a proportion of 1: 1: 2. What are the acute triangle’s angle measures?

(Hint: The sum of a triangle’s interior angles is 180o)

If we let x represent the angle measure representing the first element of the ratio. The second and third elements can be written as 2x. These three angles will have to add up to 180o.

Now, let’s find the sum of these three angles in terms of x then equate the expression to 180o. Simplify the equation to solve for x.

\begin{aligned}x + 2x + 2x &= 180\degree\\5x &= 180\degree\\x&= 36\degree\\2x&= 72\degree\end{aligned}

This means that the triangle has three acute angles measuring 36 \degree, 72 \degree, and 72 \degree. This makes it the golden triangle and the only triangle with the said ratio.

These problems show us what makes acute triangles unique. These properties and formulas will also come in handy when working on more complex problems. For now, what’s important is that you know what makes acute angles and triangles unique!