Question

CONSTRUCT ARGUMENTS Determine whether the following statement is true or false. If false, give a counterexample. If true, give an argument to support your conclusion.

If the sum of two acute angles of a triangle is greater than 90°,
then the triangle is acute.

Answer 1

The statement is true. If two acute angles of a triangle sum to more than 90°, the third angle must also be acute, making the triangle an acute triangle, as the sum of angles in any triangle must be 180°.

Step-by-step explanation:

The statement in question is: If the sum of two acute angles of a triangle is greater than 90°, then the triangle is acute. To evaluate this statement, we must understand a few basic geometric principles pertaining to triangles:

• There are three sides to a triangle.
• The sum of the angles in a triangle is 180 degrees.
• An acute triangle is one where all three angles are less than 90 degrees each.

Consider what it means if two angles of a triangle sum to more than 90°. Since the total must be 180°, the third angle must be less than 90° to satisfy this condition. Therefore, all three angles would be acute. Hence, the triangle must be acute.

We do not need a counterexample because the given statement is true. An accurate argument supporting the statement is the fact that the sum of angles in a triangle is always 180°. Given two angles adding up to more than 90°, the remaining angle must be smaller than 90° to keep the sum at 180°. Consequently, this leaves us with a triangle where all three angles are acute, defining an acute triangle.