**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.

**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.

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

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

**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.

Test your knowledge

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?

**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 $180°$.

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 $240°$.

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

Remember that the sum of angles in a triangle is equal to 180.

In an equilateral triangle, all sides and all angles are equal to each other.

Therefore, we will calculate as follows:

$x+x+x=180$

$3x=180$

We divide both sides by 3:

$x=60$

60

Choose the appropriate triangle according to the following:

Angle B equals 90 degrees.

Let's note in which of the triangles angle B forms a right angle, meaning an angle of 90 degrees.

In answers C+D, we can see that angle B is smaller than 90 degrees.

In answer A, it is equal to 90 degrees.

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

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.

Yes

In a right triangle, the sum of the two non-right angles is...?

In a right-angled triangle, there is one angle that equals 90 degrees, and the other two angles sum up to 180 degrees (sum of angles in a triangle)

Therefore, the sum of the two non-right angles is 90 degrees

$90+90=180$

90 degrees

Is the triangle in the drawing a right triangle?

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

Yes

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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