Obtuse Triangle

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Obtuse Triangle Definition

An obtuse triangle is a triangle that has one obtuse angle (greater than 90° 90° degrees and less than 180° 180° degrees) and two acute angles (each of which is less than 90° 90° degrees). The sum of all three angles together is 180° 180° degrees.

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


Exercises with Obtuse Triangles

Exercise 1

Homework:

Calculate which is larger

Given that the triangle ABC \triangle ABC is an obtuse triangle.

Which angle is larger B ∢B or A ∢A ?

Solution:

Since we are given that the triangle ABC \triangle ABC is an obtuse triangle, we understand that B∢B is not greater than 90°90°.

In a triangle, there is only one obtuse angle therefore the answer is: B>A ∢B>∢A

Answer: B>A ∢B>∢A


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Exercise 2

Given the triangle ABC \triangle ABC .

B ∢B is obtuse.

The sum of the acute angles in the triangle is equal to 70° 70° .

Find the value of angle B ∢B .

Solution:

Since we know that B ∢B is obtuse, we are certain that angles A ∢A and C ∢C are acute.

This means that we have the information that the sum of the acute angles B+A=70° ∢B+∢A=70°

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

70°+B=180° 70°+∢B=180°

B=110° ∢B=110°

Answer:

B=110° ∢B=110°


Exercise 3

Given the obtuse triangle ABC \triangle ABC .

C=12A ∢C=\frac{1}{2}∢A ,

B=3A ∢B=3∢A

Task:

Is it possible to calculate A ∢A ?

If so, calculate it.

Solution:

Given that:

C=12A ∢C=\frac{1}{2}∢A

B=3+A ∢B=3+∢A

We substitute:

A=α ∢A=α

B=3α ∢B=3α

C=12α ∢C=\frac{1}{2}α

α+3α+12α=180° α+3α+\frac{1}{2}α=180°

4.5α=180° 4.5α=180°

α=40° α=40°

Answer: yes, 40° 40° .


Do you know what the answer is?

Exercise 4

Assignment

Which triangle is given in the drawing?

Solution

Since angles ABC ABC and : ACB ACB are both equal to 70o 70^o , we know that the opposite sides are also equal, therefore the triangle is isosceles.

Answer

Isosceles triangle


Exercise 5

Assignment

Determine which of the following triangles is obtuse, which is acute, and which is right:

Solution

Let's observe triangle A A and check if it satisfies the Pythagorean theorem, therefore we replace the data we have:

52+82=92 5^2+8^2=9^2

We solve the equation

25+64=81 25+64=81

89>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 B and check if it satisfies the Pythagorean theorem, therefore we replace the data we have:

72+72=132 7^2+7^2=13^2

We solve the equation

49+49=169 49+49=169

98<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 C and check if the Pythagorean theorem is satisfied, first we calculate what is the square root of 113 113

11310.6 \sqrt{113}\approx10.6

This is the largest side among the: 3 3 and we will refer to it as "hypotenuse".

Now we replace the data we have:

72+82=1132 7^2+8^2=\sqrt{113}^2

We solve the equation

49+64=113 49+64=113

113=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


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