similarity of right triangles
Check the final answer first, then review the worked steps.
Problem
similarity of right triangles
Answer
A. True
Step-by-step solution
- Analyze the definition of similar triangles: Two triangles are considered similar if their corresponding angles are congruent and their corresponding sides are proportional. This is known as the Angle-Angle (AA) similarity criterion.
- Examine the specific case of right triangles: A right triangle has one $90^{\circ}$ angle. The sum of all angles in any triangle is $180^{\circ}$. Therefore, the two acute angles in a right triangle must sum to $180^{\circ} - 90^{\circ} = 90^{\circ}$.
- Evaluate the condition given: If the acute angles of one right triangle are congruent to the acute angles of another right triangle, then all three pairs of corresponding angles are congruent (the two acute pairs plus the pair of $90^{\circ}$ angles). According to the AA similarity postulate, if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. Since the right angle is already shared, having the acute angles congruent is sufficient to guarantee similarity.