Acute triangle | Tutorela

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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Test yourself on types of triangles!

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:

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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Test your knowledge

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?

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

examples.example_title

Given an equilateral triangle:

The perimeter of the triangle is 33 cm, what is the value of X?

examples.explanation_title

We know that in an equilateral triangle all sides are equal,

Therefore, if we know that one side is equal to X, all sides are equal to X.

We know that the perimeter of the triangle is 33.

The perimeter of the triangle is equal to the sum of the sides together.

We replace the data:

$x+x+x=33$

$3x=33$

We divide the two sections by 3:

$\frac{3x}{3}=\frac{33}{3}$

$x=11$

examples.solution_title

examples.example_title

ABCD is a square with AC as its diagonal.

What kind of triangles are ABC and ACD?

(There may be more than one correct answer!)

examples.explanation_title

Since ABCD is a square, all its angles measure 90 degrees.

Therefore, angles D and B are equal to 90°, that is, they are right angles,

Therefore, the two triangles ABC and ADC are right triangles.

In a square all sides are equal, therefore:

$AB=BC=CD=DA$

But the diagonal AC is not equal to them.

Therefore, the two previous triangles are isosceles:

$AD=DC$

$AB=BC$

examples.solution_title

Right triangles

Do you know what the answer is?

Question 1

Is the triangle in the diagram isosceles?

Question 2

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

Question 3

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

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