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Prove that two right triangles are congruent if the corresponding altitudes and angle bisectors through the right angles are congruent.
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Let ABC and A'B'C' are two right triangles with right angles C and C', respectively.
One of them (ABC) is shown in the Figure below. Regarding another triangle, please imagine it in your mind.
The angle bisector, the altitude and the median of a right triangle
CD is the altitude and CE is the angle bisector drawn from the right angle C.
Correspondingly, imagine the altitude C'D' and the angle bisector C'E' drawn from the right angle C'.
Draw the medians CF and C'F' from the right angle vertex.
In the proof, I will use this property:
In a right triangle, the right angle bisector also bisects the angle between
the altitude and the median drawn from the same vertex to the hypotenuse.
Regarding this property, see the lesson
Angle bisector drawn to the hypotenuse of a right triangle
in this site.
Since CD is congruent to C'D' and CE is congruent to C'E', the right angled triangles CDE and C'D'E' are congruent.
Hence, their corresponding angles DCE and D'C'E' are congruent.
It implies that the angles DCF and D'C'F' are congruent, since they are doubled angles DCE and D'C'E'.
Then the triangles DCE and D'C'F' are congruent, as they are right angled triangles having the pair of congruent legs DC and D'C'
and a pair of congruent acute angles.
It implies that their hypotenuses DF and D'F' are congruent.
But these hypotenuses are MEDIANS in triangles ABC and A'B'C'.
In right angled triangle, median drawn to the hypotenuse is half the length of the hypotenuse.
See the lesson
Median drawn to the hypotenuse of a right triangle
in this site.
Thus we proved that the hypotenuses AB and A'B' in our given right angled triangles are congruent.
It is just enough to state that the triangles ABC and A'B'C' are congruent.
