Obtuse Triangle

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.

Detailed explanation

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Test yourself on types of triangles!

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

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

Examples of obtuse triangles

Exercises with 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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Test your knowledge

Question 1

In an isosceles triangle, what are each of the two equal sides called ?

Question 2

In a right triangle, the two sides that form a right angle are called...?

Question 3

Does the diagram show an obtuse triangle?

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 $180°$ .

$70°+∢B=180°$

$∢B=110°$

Answer:

$∢B=110°$

Exercise 3

Given the obtuse triangle $\triangle ABC$ .

$∢C=\frac{1}{2}∢A$ ,

$∢B=3∢A$

Task:

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?

Question 1

Does the diagram show an obtuse triangle?

Question 2

Given the values of the sides of a triangle, is it a triangle with different sides?

Question 3

Given the values of the sides of a triangle, is it a triangle with different sides?

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.

Answer

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.

Answer

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

Check your understanding

Question 1

Is the triangle in the drawing an acute-angled triangle?

Question 2

Is the triangle in the drawing an acute-angled triangle?

Question 3

Is the triangle in the drawing an acute-angled triangle?

Examples with solutions for Obtuse Triangle

Exercise #1

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

Step-by-Step Solution

The problem requires us to identify the side of a right triangle that is opposite to its right angle.
In right triangles, one of the most crucial elements to recognize is the presence of a right angle (90 degrees).
The side that is directly across or opposite the right angle is known as the hypotenuse. It is also the longest side of a right triangle.
Therefore, when asked for the side opposite the right angle in a right triangle, the correct term is the hypotenuse.

Selection from the given choices corroborates our analysis:

Choice 1: Leg - In the context of right triangles, the "legs" are the two sides that form the right angle, not the side opposite to it.
Choice 2: Hypotenuse - This is the correct identification for the side opposite the right angle.

Therefore, the correct answer is $\text{Hypotenuse}$ .

Answer

Hypotenuse

Exercise #2

In an isosceles triangle, what are each of the two equal sides called ?

Step-by-Step Solution

In an isosceles triangle, there are three sides: two sides of equal length and one distinct side. Our task is to identify what the equal sides are called.

To address this, let's review the basic properties of an isosceles triangle:

An isosceles triangle is defined as a triangle with at least two sides of equal length.
The side that is different in length from the other two is usually called the "base" of the triangle.
The two equal sides of an isosceles triangle are referred to as the "legs."

Therefore, each of the two equal sides in an isosceles triangle is called a "leg."

In our problem, we confirm that the correct terminology for these two equal sides is indeed "legs," distinguishing them from the "base," which is the unequal side. This aligns with both the typical definitions and properties of an isosceles triangle.

Thus, the equal sides in an isosceles triangle are known as legs.

Answer

Legs

Exercise #3

In a right triangle, the two sides that form a right angle are called...?

Step-by-Step Solution

In a right triangle, there are specific terms for the sides. The two sides that form the right angle are referred to as the legs of the triangle. To differentiate, the side opposite the right angle is called the hypotenuse, which is distinct due to being the longest side. Hence, in response to the problem, the sides forming the right angle are correctly identified as Legs.

Answer

Legs

Exercise #4

Does the diagram show an obtuse triangle?

Video Solution

Step-by-Step Solution

To determine if the triangle in the diagram is obtuse, we will visually assess the angles:

Step 1: Identify the angles in the diagram. The triangle has three angles, with one angle appearing between the horizontal base and the left slanted side.
Step 2: Evaluate the angle between the base and the left side. If it opens wider than a right angle, it's considered obtuse. This angle seems to be greater than $90^\circ$ , indicating obtuseness.
Step 3: Conclude based on visual inspection. Since this key angle is greater than $90^\circ$ , the triangle must be an obtuse triangle.

Therefore, the solution to the problem is Yes; the diagram does show an obtuse triangle.

Answer

Yes

Exercise #5

Does the diagram show an obtuse triangle?

Video Solution

Step-by-Step Solution

To determine if the triangle shown in the diagram is obtuse, we proceed as follows:

Step 1: Identify that the diagram is indeed a triangle by observing the confluence of three edges forming a closed shape.
Step 2: Appreciate the geometric arrangement of the triangle, focusing on the sides' lengths and angles visually.
Step 3: Noticeably, the longest side of the triangle represents a noticeable tilt indicating the presence of an obtuse angle.

Based on the observation above, notably from the triangle's longest side against the base, it's clear that one angle is larger than $90^\circ$ . Hence, the triangle in the diagram is indeed an obtuse triangle.

Therefore, the correct answer is Yes.

Answer

Yes

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Obtuse Triangle

Obtuse Triangle Definition

Test yourself on types of triangles!

Obtuse triangle

Examples of obtuse triangles

Exercises with Obtuse Triangles

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Examples with solutions for Obtuse Triangle

Exercise #1

Step-by-Step Solution

Answer

Exercise #2

Step-by-Step Solution

Answer

Exercise #3

Step-by-Step Solution

Answer

Exercise #4

Video Solution

Step-by-Step Solution

Answer

Exercise #5

Video Solution

Step-by-Step Solution

Answer

More Questions

Types of Triangles