Can an Obtuse Triangle be Isoseles?

Yes, an obtuse triangle can also be isosceles. To understand why, let's first define the key terms and explore some examples.

Definition of Obtuse and Isosceles Triangles

An obtuse triangle is a triangle where one of its internal angles is greater than 90 degrees. An isosceles triangle is a triangle that has at least two sides of equal length, which also means that it has at least two angles that are equal.

Conditions for an Obtuse Isosceles Triangle

For a triangle to be both obtuse and isosceles, it must meet two criteria: it must have one angle greater than 90 degrees, and it must have at least two sides of equal length.

Example 1: 70-70-40 Triangle

Consider an isosceles triangle where each of the two equal angles is 70 degrees. The third angle would then be:

180 - 70 - 70 40 degrees

This triangle is acute, not obtuse. So, in this case, the triangle is not obtuse.

Example 2: 120-30-30 Triangle

For a triangle to be both obtuse and isosceles, the two equal angles must be such that the sum of all three angles is 180 degrees and one of the angles must be greater than 90 degrees. For example, consider a triangle where two angles are 30 degrees, and one angle is 120 degrees.

30 30 120 180 degrees

In this case, the triangle is both obtuse (since one angle is 120 degrees) and isosceles (since the two 30-degree angles are equal).

Additional Examples

1. **Three Angles Acute**: An isosceles triangle can have all three angles acute. For instance, a triangle with angles 50-70-60 is isosceles (since 70-70-40 would not be obtuse) and acute.

2. **One Right Angle**: An isosceles triangle can have one right angle and the other two angles equal and each 45 degrees. For example, a triangle with angles 90-45-45 is isosceles and has one right angle.

3. **One Obtuse Angle and Two Smaller Equal Angles**: An isosceles triangle can have one angle obtuse, and the other two angles equal but less than 45 degrees. For instance, a triangle with angles 120-30-30 is isosceles and has one obtuse angle.

Conclusion

As demonstrated through the examples, an obtuse triangle can indeed be isosceles, provided the conditions that one angle is greater than 90 degrees and at least two sides of the triangle are of equal length.