Hypotenuse-Leg Theorem

key notes :

Definition

  • The Hypotenuse-Leg (HL) Theorem is a rule used to prove that two right triangles are congruent.
  • It states: If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the two triangles are congruent.

Conditions for HL Theorem

  • The triangles must be right triangles (have a 90° angle).
  • The hypotenuse (longest side) of both triangles must be congruent.
  • One leg (shorter side) of both triangles must be congruent.

Why HL Theorem Works

  • The Pythagorean Theorem ensures that if two sides of a right triangle are known, the third side is uniquely determined.
  • Unlike SSA (Side-Side-Angle), which is not a valid congruence rule, HL is valid because of the right angle constraint.

Comparison with Other Triangle Congruence Theorems

HL is only applicable to right triangles.

Other congruence postulates for any triangles:

  • SSS (Side-Side-Side)
  • SAS (Side-Angle-Side)
  • ASA (Angle-Side-Angle)
  • AAS (Angle-Angle-Side)

Examples

  • If two right triangles have hypotenuses of 10 cm and one leg of 6 cm, they must be congruent by HL Theorem.
  • Used in proving properties of right triangles and isosceles right triangles.

Application in Real Life

  • Engineering and construction use HL for measuring distances and verifying right angles.
  • Architecture relies on HL for ensuring symmetrical designs.

Learn with an example

What values of v and w make △FGH ≅ △DCE?

v = _____

w = _____

Use the Hypotenuse-Leg Theorem to solve for the values of v and w that make the triangles congruent.

First, look at the labelled pair of corresponding legs. The first leg FH has a length of v and the second leg DE has a length of 18. For the triangles to be congruent, the leg lengths must be equal. So, v must equal 18.

Now, look at the hypotenuses. The first hypotenuse FG has a length of 42 and the second hypotenuse DC has a length of w. For the triangles to be congruent, the hypotenuse lengths must be equal. So, w must equal 42.

The values of v and w that make △FGH ≅ △DCE are v=18 and w=42.

What values of x and y make △GHI ≅ △EDF?

X =

Y =

Use the Hypotenuse-Leg Theorem to solve for the values of x and y that make the triangles congruent.

First, look at the labelled pair of corresponding legs. The first leg HI has a length of x and the second leg DF has a length of 44. For the triangles to be congruent, the leg lengths must be equal. So, x must equal 44.

Now, look at the hypotenuses. The first hypotenuse GI has a length of 49 and the second hypotenuse EF has a length of y. For the triangles to be congruent, the hypotenuse lengths must be equal. So, y must equal 49.

The values of x and y that make △GHI≅△EDF are x=44 and y=49.

Let’s try some problems!✍️