**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.

**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.

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

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

**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:** $∢B>∢A$

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?

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:**

$∢B=110°$

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

Does the diagram show an obtuse triangle?

Question 3

Does the diagram show an obtuse triangle?

**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

**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

Does the diagram show an obtuse triangle?

Question 2

Fill in the blanks:

In an isosceles triangle, the angle between two ___ is called the "___ angle".

Question 3

Given the size of the 3 sides of the triangle, is it an equilateral triangle?

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

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