Acute triangle

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.

Detailed explanation

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Calculate the size of angle X given that the triangle is equilateral.

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

3 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.

Answer:

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

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Question 1

Can a right triangle be equilateral?

Question 2

Choose the appropriate triangle according to the following:

Angle B equals 90 degrees.

Question 3

Does every right triangle have an angle _____ The other two angles are _______

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 $90°$

Task:

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.

Answer:

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 $35°$

Task:

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.

Answer:

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

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

Video Solution

Step-by-Step Solution

An acute-angled triangle is defined as a triangle where all three interior angles are less than $90^\circ$ .

In examining the visual depiction of the triangle provided in the problem, we need to see if it appears to satisfy this property. The assessment relies on observing the triangle's structure shown in the drawing and noting any geometric indications suggesting angle types.

Given the information from the drawing, if all angles seem to satisfy the condition of being less than $90^\circ$ , then by definition, the triangle is an acute-angled triangle.

Conclusively, the answer to whether the triangle is acute-angled based on provided visual assessment and inherent assumptions in its illustration is: Yes.

Answer

Yes

Exercise #2

In an isosceles triangle, the angle between ? and ? is the "base angle".

Step-by-Step Solution

An isosceles triangle is one that has at least two sides of equal length. The angles opposite these two sides are known as the "base angles."
The side that is not equal to the other two is referred to as the "base" of the triangle. Thus, the "base angles" are the angles between each of the sides that are equal in length and the base.
Therefore, when we specify the angle in terms of its location or position, it is the angle between a "side" and the "base." This leads to the conclusion that the angle between the side and the base is the "base angle."

Therefore, the correct choice is Side, base.

Answer

Side, base.

Exercise #3

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

Video Solution

Step-by-Step Solution

As is known, a scalene triangle is a triangle in which each side has a different length.

According to the given information, this is indeed a triangle where each side has a different length.

Answer

Yes

Exercise #4

Is the triangle in the drawing a right triangle?

Step-by-Step Solution

Due to the presence of the 90 degree angle symbol we can determine that this is indeed a right-angled triangle.

Answer

Yes

Exercise #5

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

Do you know what the answer is?

Question 1

Does the diagram show an obtuse triangle?

Question 2

Does the diagram show an obtuse triangle?

Question 3

Does the diagram show an obtuse triangle?

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Acute triangle

Definition of Acute Triangle

Test yourself on types of triangles!

Acute triangle

3 Examples of acute triangles

Exercises with Acute Triangles

Exercise 1

Exercise 2

Exercise 3

Examples and exercises with solutions for acute triangles

Exercise #1

Video Solution

Step-by-Step Solution

Answer

Exercise #2

Step-by-Step Solution

Answer

Exercise #3

Video Solution

Step-by-Step Solution

Answer

Exercise #4

Step-by-Step Solution

Answer

Exercise #5

Step-by-Step Solution

Answer

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Types of Triangles