For right triangles, special congruence conditions can be used in addition to the general congruence conditions.
Congruence Conditions for Right Triangles
- The hypotenuse and one acute angle are respectively equal
- The hypotenuse and one other side are respectively equal
Example ①
When ∠C=∠F=90°, AB=DE, and BC=EF, prove that △ABC≅△DEF.
Answer: In △ABC and △DEF / ∠C=∠F=90° (right angles) ① / AB=DE (hypotenuses) ② / BC=EF (one other side) ③ / Since the hypotenuse and one other side of the right triangles are respectively equal, △ABC≅△DEF.
When ∠C=∠F=90°, AB=DE, and BC=EF, prove that △ABC≅△DEF.
Answer: In △ABC and △DEF / ∠C=∠F=90° (right angles) ① / AB=DE (hypotenuses) ② / BC=EF (one other side) ③ / Since the hypotenuse and one other side of the right triangles are respectively equal, △ABC≅△DEF.
Key Points
- There are two congruence conditions for right triangles (hypotenuse + one acute angle, hypotenuse + one other side)
- The hypotenuse must always be included
Practice Problem
- When ∠B=∠E=90°, AC=DF, and ∠A=∠D, state the congruence condition that can be used.
Answers and Explanations
- Answer: The hypotenuses of the right triangles (AC=DF) and one acute angle (∠A=∠D) are respectively equal.