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Congruence in Right
Triangles
Two sides and a non-included angle of one
triangle are congruent to two sides and the
non-included angle of another triangle.
Notice the two triangles are not congruent.
Side-Side-Angle is not a valid method for
proving two triangles congruent.
However, this method works in the special
case of right triangles where the right
angles are the non-included angles.
Hypotenuse
Right Triangle
Leg
Leg
Right Triangle – A triangle with one right
angle. (90° angle).
Legs – The sides that form the right angle.
Hypotenuse – The side opposite the right
angle. The longest side of the triangle.
Hypotenuse-Leg (HL) Theorem
If the hypotenuse and leg of one right
triangle are congruent to the hypotenuse
and leg of another right triangle, then the
triangles are congruent.
X
Proof of HL Theorem
Given: XR  XZ , XY  XY , XY  RZ
Prove: XRY  XZY
R
Y
Z
Statements
Reasons
1. XR  XZ , XY  XY , XY  RZ
1. Given
2. XYR is a right angle,
2. Perpendicular lines form right angles. (1)
XYZ is a right angle
3. XYR  XYZ
4. R  Z
3. All right angles are congruent. (2)
5. XRY  XZY
5. AAS (1, 3, 4)
4. If two sides of a triangle are congruent,
the angles opposite are congruent. (1)