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° 90° degrees and the sum of all three together equals 180° 180° degrees. 

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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:


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 5²+8²=9²

25+64=81 25+64=81

89>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 7²+7²=13²

49+49=169 49+49=169

169>98 169>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.

10.6113 10.6≈\sqrt{113}

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

72+82=1132 7²+8²=\sqrt{113}²

49+64=113 49+64=113

113=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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Exercise 2

Let's look at 3 angles

Angle A is equal to 30° 30°

Angle B is equal to 60° 60°

Angle C is equal to 90° 90°

Task:

Can these angles form a triangle?

Solution:

30+60+90=180 30+60+90=180

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

therefore these angles can form a triangle.

Answer:

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


Exercise 3

Angle A is equal to 90° 90°

Angle B is equal to 115° 115°

Angle C is equal to 35° 35°

Task:

Can these angles form a triangle?

Solution:

90°+115°+35°=240° 90°+115°+35°=240°

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

therefore these angles cannot form a triangle.

Answer:

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


Examples and exercises with solutions for acute triangles

examples.example_title

Given an equilateral triangle:

XXX

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 x+x+x=33

3x=33 3x=33

We divide the two sections by 3:

3x3=333 \frac{3x}{3}=\frac{33}{3}

x=11 x=11

examples.solution_title

11

examples.example_title

AAABBBCCCDDD

ABCD is a square, and a diagonal AC is drawn there.

How can we define the triangles ABC and ACD?

(Attention! 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 AB=BC=CD=DA

But the diagonal AC is not equal to them.

Therefore, the two previous triangles are isosceles:

AD=DC AD=DC

AB=BC AB=BC

examples.solution_title

Right triangles

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