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Hypotenuse Leg Theorem

In a right-angled triangle, the hypotenuse is the longest side which is always opposite to the right angle. The hypotenuse leg theorem states that two right triangles are congruent if the hypotenuse and one leg of one right triangle are congruent to the other right triangle’s hypotenuse and leg side. In order to prove any two right triangles congruent, we apply the HL (Hypotenuse Leg) Theorem or the RHS (Right angle-Hypotenuse-Side) congruence rule. Let us learn more about the hypotenuse leg theorem in this page.

1. Hypotenuse Leg Theorem
2. Hypotenuse Leg Theorem Proof
3. Solved Examples on Hypotenuse Leg Theorem
4. Practice Questions on Hypotenuse Leg Theorem
5. FAQs on Hypotenuse Leg Theorem

Hypotenuse Leg Theorem

According to the hypotenuse leg theorem, if the hypotenuse and one leg of one right triangle are congruent to the other right triangle’s hypotenuse and leg side, then the two triangles are congruent. In other words, a given set of right triangles are congruent if the corresponding lengths of their hypotenuse and one leg are equal. The hypotenuse leg theorem is a criterion that is used to prove the congruence of triangles. In the other congruency postulates like, Side Side Side (SSS), Side Angle Side (SAS), Angle Side Angle (ASA), and Angle Angle Side (AAS), three criteria are tested, whereas, in the hypotenuse leg (HL) theorem, only the hypotenuse and one leg are considered. Observe the following figure which shows a right-angled triangle with two perpendicular legs and a hypotenuse.

Hypotenuse Leg Theorem Proof

The proof of the hypotenuse leg theorem shows how a given set of right triangles are congruent if the corresponding lengths of their hypotenuse and one leg are equal. Observe the following isosceles triangle ABC in which side AB = AC and AD is perpendicular to BC.

Given: Here, ABC is an isosceles triangle, AB = AC, and AD is perpendicular to BC.
Proof:
AD, being an altitude is perpendicular to BC and forms ADB and ADC as right-angled triangles. AB and AC are the respective hypotenuses of these triangles, and we know they are equal to each other. AD = AD because they are common in both the triangles.
So, AB = AC and AD is common.
Therefore, a hypotenuse and a leg pair in two right triangles, are satisfying the definition of the HL theorem.
We know that angles B and C are equal (Isosceles Triangle Property).
We also know that the angles BAD and CAD are equal.(AD bisects BC, which makes BD equal to CD).
Therefore, △ADB ≅ △ADC
Hence proved.

Important Notes

  • The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides (base and perpendicular). This is represented as: Hypotenuse² = Base² + Perpendicular².
  • According to the HL Congruence rule, the hypotenuse and one leg are the elements that are used to test the congruence of triangles.
  • The HL Congruence rule is similar to the SAS (Side-Angle-Side) postulate. The only difference is that SAS needs two sides and the included angle, whereas, in the HL theorem, the known angle is the right angle, which is not the included angle between the hypotenuse and the leg.

Related Articles on Hypotenuse Leg Theorem

Check out the following pages related to the hypotenuse leg theorem.

Solved Examples on Hypotenuse Leg Theorem

  1. Example 1. If △ABC ≅ △PQR, what is the value of x and y?

    Solution:

    Following the HL theorem, in △ABC and △PQR: BC = QR (congruent hypotenuse)

    Thus, y = 13
    AC = PQ (congruent legs)
    Thus, x = 5.
    Therefore, x = 13, y = 5.

  2. Example 2. Fred wondered if the Hypotenuse Leg Theorem can be proved using the Pythagorean theorem. Can you find out?

    Solution:

    In the figure given above, triangles ABC and XYZ are right triangles with AB = YZ, AC = XZ.

    By Pythagorean Theorem,

    (AC)² = (AB)² + (BC)² and (XZ)² = (XY)² + (YZ)²

    Since AC = XZ, we can write that: (AB)² + (BC)² = (XY)² + (YZ)²—> (Equation 1)
    It is given that AB = YZ,
    Substituting AB with YZ in Equation 1:
    (YZ)² + (BC)² = (XY)² + (YZ)²
    Solving the equation: we get (BC)² = (XY)². This means side BC = XY. Hence, △ABC ≅ △XYZ. Thus, with the help of the Pythagorean theorem, the Hypotenuse leg theorem was proved, which says that if the hypotenuse and one leg of one right triangle are congruent to the other right triangle’s hypotenuse and leg side, then the two triangles are congruent.

  3. Example 3. For the given figure, prove that △PSR ≅ △PQR.

    Solution:

    It is given that △PSR and △PQR are right-angled triangles.

    PS = QR (equal legs, given)
    PR = PR (equal and common hypotenuse)
    Hence, △PSR ≅ △PQR (by HL rule)

Practice Questions on Hypotenuse Leg Theorem

FAQs on Hypotenuse Leg Theorem

What is the Difference Between the Legs and the Hypotenuse of a Triangle?

In a right-angled triangle, the side opposite to the right angle is called the hypotenuse and the two other adjacent sides are called its legs. The hypotenuse is the longest side of the triangle, while the other two legs are always shorter in length.

What is the Formula to Calculate the Hypotenuse of a Right-Angled Triangle?

To calculate the hypotenuse of a right-angled triangle we use the Pythagoraean Theorem: Hypotenuse = √(Base2 + Perpendicular2).

What is the Hypotenuse Leg Theorem?

The hypotenuse leg theorem states that if the hypotenuse and one leg of one right triangle are congruent to the other right triangle’s hypotenuse and leg side, then the two triangles are congruent.

What is the Pythagorean Theorem?

According to the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This is represented as: Hypotenuse² = Base² + Perpendicular².

What is the SSA Theorem?

SSA (Side-Side-Angle) refers to one of the criteria for the congruence of two triangles. It is justified when the two sides and an angle (not included between them) of a triangle are respectively equal to two sides and an angle of another triangle.

What is the Use of the Pythagoras Theorem?

The Pythagoras theorem works only for right-angled triangles and follows the rule: Hypotenuse² = Base² + Perpendicular². When any two values are known, we can apply the theorem and calculate the missing values.

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