# Acute triangle

🏆Practice types of triangles

## Definition of Acute Triangle

An acute triangle has all acute angles, meaning each of its three angles measures less than $90°$ degrees and the sum of all three together equals $180°$ degrees.

## Test yourself on types of triangles!

In a right triangle, the side opposite the right angle is called....?

Next, we will look at some examples of acute triangles:

## Exercises with Acute Triangles

### Exercise 1

Assignment:

Determine which of the following triangles is obtuse, which is acute, and which is a right triangle:

Solution:

A. We will examine if the Pythagorean theorem holds for this triangle:

$5²+8²=9²$

$25+64=81$

$89>81$

The sum of the squares of the perpendicular sides is greater than the square of the remaining side, therefore it is an acute-angled triangle.

B. Now we will examine this triangle:

$7²+7²=13²$

$49+49=169$

$169>98$

The sum of the squares of the perpendicular sides is greater than the square of the remaining side, therefore it is an obtuse-angled triangle.

$10.6≈\sqrt{113}$

C. The longest side of the 3 will be treated as the hypotenuse.

$7²+8²=\sqrt{113}²$

$49+64=113$

$113=113$

The Pythagorean theorem holds true and therefore triangle 3 is a right triangle.

A-acute angle acute B-obtuse angle obtuse C-right angle right.

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### Exercise 2

Let's look at 3 angles

Angle A is equal to $30°$

Angle B is equal to $60°$

Angle C is equal to $90°$

Can these angles form a triangle?

Solution:

$30+60+90=180$

The sum of the angles in a triangle is equal to $180°$,

therefore these angles can form a triangle.

Yes, since the sum of the internal angles of a triangle is equal to $180°$.

### Exercise 3

Angle A is equal to $90°$

Angle B is equal to $115°$

Angle C is equal to $35°$

Can these angles form a triangle?

Solution:

$90°+115°+35°=240°$

The sum of the angles is greater than $180°$,

therefore these angles cannot form a triangle.

No, since the sum of the internal angles must be $180°$, and in this case the angles add up to $240°$.

## Examples and exercises with solutions for acute triangles

### Exercise #1

What kind of triangle is given in the drawing?

### Step-by-Step Solution

As all the angles of a triangle are less than 90° and the sum of the angles of a triangle equals 180°:

$70+70+40=180$

The triangle is isosceles.

Isosceles triangle

### Exercise #2

Which kind of triangle is given in the drawing?

### Step-by-Step Solution

As we know that sides AB, BC, and CA are all equal to 6,

All are equal to each other and, therefore, the triangle is equilateral.

Equilateral triangle

### Exercise #3

What kid of triangle is the following

### Step-by-Step Solution

Given that in an obtuse triangle it is enough for one of the angles to be greater than 90°, and in the given triangle we have an angle C greater than 90°,

$C=107$

Furthermore, the sum of the angles of the given triangle is 180 degrees so it is indeed a triangle:

$107+34+39=180$

The triangle is obtuse.

Obtuse Triangle

### Exercise #4

What kind of triangle is given in the drawing?

### Step-by-Step Solution

Given that sides AB and AC are both equal to 9, which means that the legs of the triangle are equal and the base BC is equal to 5,

Therefore, the triangle is isosceles.

Isosceles triangle

### Exercise #5

What kind of triangle is given here?

### Step-by-Step Solution

Since none of the sides have the same length, it is a scalene triangle.