# Obtuse Triangle

🏆Practice types of triangles

## Obtuse Triangle Definition

An obtuse triangle is a triangle that has one obtuse angle (greater than $90°$ degrees and less than $180°$ degrees) and two acute angles (each of which is less than $90°$ degrees). The sum of all three angles together is $180°$ degrees.

## Test yourself on types of triangles!

What kid of triangle is given in the drawing?

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

## Exercise 1

Homework:

Calculate which is larger

Given that the triangle $\triangle ABC$ is an obtuse triangle.

Which angle is larger $∢B$ or $∢A$?

Solution:

Since we are given that the triangle $\triangle ABC$ is an obtuse triangle, we understand that $∢B$ is not greater than $90°$.

In a triangle, there is only one obtuse angle therefore the answer is: $∢B>∢A$

Answer: $∢B>∢A$

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### Exercise 2

Given the triangle $\triangle ABC$.

$∢B$ is obtuse.

The sum of the acute angles in the triangle is equal to $70°$.

Find the value of angle $∢B$.

Solution:

Since we know that $∢B$ is obtuse, we are certain that angles $∢A$ and $∢C$ are acute.

This means that we have the information that the sum of the acute angles $∢B+∢A=70°$

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

$70°+∢B=180°$

$∢B=110°$

$∢B=110°$

### Exercise 3

Given the obtuse triangle $\triangle ABC$.

$∢C=\frac{1}{2}∢A$,

$∢B=3∢A$

Is it possible to calculate $∢A$?

If so, calculate it.

Solution:

Given that:

$∢C=\frac{1}{2}∢A$

$∢B=3+∢A$

We substitute:

$∢A=α$

$∢B=3α$

$∢C=\frac{1}{2}α$

$α+3α+\frac{1}{2}α=180°$

$4.5α=180°$

$α=40°$

Answer: yes, $40°$.

Do you know what the answer is?

### Exercise 4

Assignment

Which triangle is given in the drawing?

Solution

Since angles $ABC$ and : $ACB$ are both equal to $70^o$, we know that the opposite sides are also equal, therefore the triangle is isosceles.

Isosceles triangle

### Exercise 5

Assignment

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

Solution

Let's observe triangle $A$ and check if it satisfies the Pythagorean theorem, therefore we replace the data we have:

$5^2+8^2=9^2$

We solve the equation

$25+64=81$

$89>81$

The sum of the squares of the "perpendicular" is greater than the square of the rest, therefore the triangle is an isosceles triangle.

Let's observe triangle $B$ and check if it satisfies the Pythagorean theorem, therefore we replace the data we have:

$7^2+7^2=13^2$

We solve the equation

$49+49=169$

$98<169$

The sum of the squares of the "perpendicular" is less than the square of the other, therefore the triangle is obtuse

Let's observe triangle $C$ and check if the Pythagorean theorem is satisfied, first we calculate what is the square root of $113$

$\sqrt{113}\approx10.6$

This is the largest side among the: $3$ and we will refer to it as "hypotenuse".

Now we replace the data we have:

$7^2+8^2=\sqrt{113}^2$

We solve the equation

$49+64=113$

$113=113$

In this triangle, the Pythagorean theorem is satisfied and therefore the triangle is right.

A: acute angle B: obtuse angle C: right angle

## examples with solutions for obtuse triangle

### Exercise #1

What kid of triangle is given in the drawing?

### Step-by-Step Solution

The measure of angle C is 90°, therefore it is a right angle.

If one of the angles of the triangle is right, it is a right triangle.

Right triangle

### Exercise #2

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 #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

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

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