Explain why a triangle cannot have more than one obtuse angle.

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durbanville | High School Teacher | (Level 2) Educator Emeritus

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In Geometry, there are various possible angle sizes and each angle falls into the category of either an acute angle (smaller than 90 degrees), a right angle (90 degrees), an obtuse angle (bigger than 90 degrees but smaller than 180 degrees), a straight angle (180 degrees), a reflex angle (bigger than 180 degrees but smaller than 360 degrees) and a revolution (360 degrees).

In flat or traditional Geometry, the properties of a triangle of any size mean that a triangle can never have more than 180 degrees between all its angles. It follows therefore that, in flat Geometry, a triangle cannot have a revolution, which at 360 degrees is a full circle around a point. A triangle cannot have a reflex angle either as this is an angle bigger than 180 degrees and so the reflex angle would prevent the joining of the three points of a triangle. Do note that a triangle can be formed with the reflex angle on the outside. Picture a triangle. The angle around any of the three corners or vertices (i.e. around the outside) could be reflex. In such a case, the angle on the inside could be either acute or obtuse, depending on the triangle. 

Two obtuse angles by definition mean that there would be two angles of (at least) 91 degrees each. 91 + 91 = 182 degrees. That is already too much for a triangle, even before the third angle has been considered. Therefore, a triangle can never have more than one obtuse angle.

When an angle of a triangle is 90 degrees, the triangle cannot have an obtuse angle. The other two must each be less than 90 degrees (90 deg + 89 deg + 1 deg = 180 deg). Note that the other two angles can be any combination but both would be acute angles (smaller than 90 degrees). 

Therefore, in any triangle where there is at least one obtuse angle (greater than 90 degrees but smaller than 180 degrees), it means that the triangle can only have its two other angles smaller than 90 degrees because an obtuse angle (for example, 91 degrees) means there is only 89 degrees (at the most) left for the other two angles. It therefore follows that they must both be less than 90 degrees and so must both be acute. 

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Jyotsana | Student, Grade 10 | (Level 1) Valedictorian

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Triangle cannot have more than one obtuse angle because there are 3 side and if you add three sides it equals to 180 degree and obtuse angle is greater than 90 degree and the other two has to be less than 90 degree.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll consider the three angles of triangle as: A,B,C.

By definition, the sum of angles of a triangle is 180 degrees.

A+B+C = 180 degrees

We'll consider A as being the obtuse angle of the triangle ABC.

Therefore A>90 degrees.

Since A>90 degrees, then B+C < 90.

This tells us that B and C are acute angles.

Therefore, a triangle cannot have more than one obtuse angle.

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