Properties of Triangles

Congruence Conditions for Right Triangles

Two right triangles are congruent in the following cases:
  1. If the lengths of the hypotenuse and one acute angle are equal ( congruence).
  2. If the lengths of the hypotenuse and one other side are equal ( congruence).

[Explanation]

i. Consider two right triangles and with , as shown in the diagram:
Since ,
From (1), (2), and (3), by the ASA congruence theorem.
Thus, two right triangles are congruent if the hypotenuse and one acute angle are equal.
ii. Consider two right triangles and with , where
When the equal sides and are aligned, the sum of the angles at . This means the points , , and are collinear.
From the triangle is isosceles, so
Using (1) and (2), the two right triangles have the same hypotenuse length and one other side of equal length. Therefore,
Thus, two right triangles are congruent if the hypotenuse and one other side are equal.