First of all, an isosceles triangle has two sides, the legs, that are congruent. Second, an acute triangle is a triangle where all of the angles are less than 90∘. Thus, if one angle is greater than or equal to 90∘, its no longer an acute triangle. Consider the following acute isosceles triangle.
Can we change one angle from acute to a right or obtuse angle by extending one or more sides?
Extending the legs
If we extend the legs, we have to extend both of them by the same amount to make sure the triangle is still isosceles. As we can see from the diagram below, when you extend the legs, the base angles will approach right angles and the vertex angle will approach 0∘.
Note that the base angles can never exceed or equal 90∘ as the legs would then never meet up to form a triangle.
Therefore, extending the legs will never cause the isosceles triangle to become right or obtuse.
Extending the base
If we extend the base, the base angles will become smaller and smaller, which means the vertex angle becomes larger.
If we extend the base far enough, the vertex angle can go from an acute angle to a right angle to an obtuse angle.
Conclusion
An isosceles triangle is not always an acute triangle.
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