Question

1. it is impossible for a triangle to be which of the following?

○ both right and isosceles
○ both obtuse and acute
○ both acute and equilateral
○ both equiangular and equilateral
○ all of these triangles are possible.

Explanation:

Step1: Recall triangle angle sum rule

The sum of interior angles of a triangle is $180^\circ$.

Step2: Analyze each option

• Right and isosceles: A triangle with angles $90^\circ, 45^\circ, 45^\circ$ is valid, so this is possible.
• Obtuse and acute: An obtuse angle is $>90^\circ$, acute is $<90^\circ$. A triangle with $100^\circ, 50^\circ, 30^\circ$ is valid, so this is possible.
• Acute and equilateral: An equilateral triangle has all $60^\circ$ angles, which are acute, so this is possible.
• Equiangular and equilateral: Equiangular triangles have all equal angles, which makes all sides equal (equilateral), so this is possible.

Answer:

All of these triangles are possible.