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Home » Triangles » Obtuse Triangle - Definition and Properties

Obtuse Triangle - Definition and Properties

Last updated: May 1, 2024 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

An obtuse triangle is a triangle in which one of the angles is larger than 90°. There is only one such angle possible, since the sum of angles in a triangle is 180°.

It's easy to show that in this type of triangle, the obtuse angle is larger than the sum of the other two angles.

It's also easy to show the converse- that if an angle in a triangle is larger than the sum of the other two, then the triangle is obtuse. Both of these follow directly from the fact that the sum of angles in a triangle is 180°.

Problem

Show that in an obtuse triangle, the obtuse angle is larger than the sum of the other two angles. Also show the converse- that if an angle in a triangle is larger than the sum of the other two, then the triangle is obtuse.

obtuse triangle

Strategy

Both of these problems are a direct consequence of the fact that the sum of all three angles in a triangle is exactly 180°. So our strategy is to prove them using arithmetic manipulation of the sum of the angles.

Proof

(1) m∠1 + m∠2 + m∠3= 180°                      //sum of angles in a triangle is 180°
(2) m∠2>90° // Given, ΔABC is obtuse
(3) m∠1 + m∠3 < 90° // subtract (2) from (1)
(4) m∠1 + m∠3 < m∠2 //(2), (3)

Now lets do the converse, show that if m∠1 + m∠3 < m∠2, then ΔABC is obtuse:

(5) m∠1 + m∠2 + m∠3= 180°                      //sum of angles in a triangle is 180°
(6) m∠1 + m∠3 < m∠2 // Given
(7) m∠2 >180° -m∠2 // subtract (6) from (5)
(8) 2 * m∠2 > 180° //Add m∠2 to both sides
(9) m∠2 > 90° //divide by 2
(10) ΔABC is obtuse //(9) , defintion of obtuse triangle

« The Scalene Inequality Theorem
Area of a Trapezoid »

About the Author

Ido Sarig is a high-tech executive with a BSc degree in Computer Engineering. His goal is to help you develop a better way to approach and solve geometry problems. You can contact him at [email protected].

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Welcome to Geometry Help! I'm Ido Sarig, a high-tech executive with a BSc degree in Computer Engineering and an MBA degree in Management of Technology. I'm here to tell you that geometry doesn't have to be so hard! My goal with this website is to help you develop a better way to approach and solve geometry problems, even if spatial awareness is not your strongest quality. Read More…

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