Question

which statement is true? a scalene triangle and a right triangle are always congruent. a scalene triangle and a right triangle are never congruent. an equilateral triangle and a right triangle are always congruent. an equilateral triangle and a right triangle are never congruent.

Explanation:

• For scalene and right - triangle:
• A scalene triangle has all sides of different lengths. A right - triangle has one angle equal to \(90^{\circ}\). A scalene triangle can be a right - triangle (e.g., sides \(3,4,5\): scalene and right). So, the statements “A scalene triangle and a right triangle are always congruent” and “A scalene triangle and a right triangle are never congruent” are false. Congruent triangles require all corresponding sides and angles to be equal. Just because a triangle is scalene and right doesn't mean all such pairs are congruent (there can be scalene right - triangles of different sizes), but they can exist (so “never” and “always” for scalene and right is wrong).
• For equilateral and right - triangle:
• An equilateral triangle has all angles equal to \(60^{\circ}\). A right - triangle has one angle equal to \(90^{\circ}\). Since the angle measures of an equilateral triangle (\(60^{\circ},60^{\circ},60^{\circ}\)) and a right - triangle (\(90^{\circ},a,b\) where \(a + b=90^{\circ}\)) can never be the same (by the angle - sum property of triangles \(A + B + C=180^{\circ}\)), an equilateral triangle and a right - triangle can never be congruent (congruent triangles have equal corresponding angles and sides).

Answer:

An equilateral triangle and a right triangle are never congruent.