Hypotenuse Leg Theorem – Definition, Proof, Examples, FAQs

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What Is the Hypotenuse Leg Theorem?

Hypotenuse Leg Theorem (HL theorem) is used to prove the congruence of two right angled triangles. It is also known as the RHS (Right angle-Hypotenuse-Side) congruence rule. 

In a right triangle, there is one right angle ($90^{\circ}$ angle) and two acute angles. The side opposite to the right angle is called hypotenuse. It is the longest side of the right triangle. The other two sides are called ‘legs’ of the right triangle.

Right triangle (legs and hypotenuse)

According to the HL theorem, if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of the other right triangle, then the two triangles are congruent.

HL theorem congruence example

Hypotenuse Leg Theorem Statement

HL triangle congruence theorem: If the hypotenuse and one leg of one right triangle are congruent to the hypotenuse and leg of the other right triangle, then the two right triangles are said to be congruent. 

In other words, two right triangles are congruent if the corresponding lengths of the hypotenuse and one leg are equal.  

Hypotenuse Leg Theorem Proof

Given: In right triangles △ABC ≅△XYZ, AB = XY and AC = XZ

Right triangles ABC and XYZ

To prove: △ABC ≅△XYZ

Proof: 

StatementReason
AB2 + BC2 = AC2Pythagoras theorem
XY2 + YZ2 = XZ2Pythagoras theorem
XZ = AC                                            …(1)Given     
AB2 + BC2 = XY2 + YZ2                                    …(2)From (1)
AB = XY                                            …(3)Given
XY2 + BC2 = XY2 + YZ2From (2) and (3)
BC2 = YZ2Canceling out XY2
BC = XYTaking positive square root
△ABC ≅△XYZBy SSS criterion

Facts about Hypotenuse Leg Theorem

  • HL theorem (Hypotenuse Leg Theorem or Hypotenuse Leg Congruence Theorem) is also known as the RHS (Right angle-Hypotenuse-Side) congruence rule.
  • The LA Theorem (Leg- Acute angle Theorem): If the leg and an acute angle of one right triangle are both congruent to the corresponding leg and acute angle of another right triangle, the two triangles are congruent.

Conclusion

In this article, we learned about HL theorem or HL congruence criterion to prove congruence of two right triangles. We learned the statement and also proved the theorem. Let’s use the HL theorem to solve a few examples and practice problems.

Solved Examples on HL Theorem (Hypotenuse Leg Theorem)

1. Is △PQR △NLM? 

Right triangles PQR and LMN

Solution: 

To prove the congruence of right triangles, we will use the HL theorem.

In right triangles △PQR and △LMN, we have

∠Q = ∠L = 90°

PR = MN = 13 units

QR = LM = 5 units

By HL Theorem, one leg and hypotenuse of △PQR are congruent to one leg and hypotenuse of △LMN.

Thus, △PQR ≅ △NLM

2. In the △PQR, if PS is the perpendicular bisector of the side QR, then prove that ∠QPS = ∠RPS.

Two right triangles with one common leg

Solution: 

In △PQR, PS is the perpendicular bisector of QR.

$\Rightarrow$ QS = SR …(1)

In △PSQ and △PSR

∠PSQ = ∠PSR = 90°

PQ = PR = 3 units

QS = SR …from (1)

Thus, △PSQ ≅ △PSR …by HL Theorem

$\Rightarrow$ ∠QPS = ∠RPS …CPCTC (Corresponding parts of congruent triangles are congruent.)

3. In the following figure, find the value of JK and MN.

Right triangles IJK and MNO

Solution: 

In △IJK and △ONM,

∠IJK = ∠ONM = 90°

IK = MO = 5 units

IJ = ON = 4 units

By HL Theorem,

△IJK ≅ △ONM

$\Rightarrow$KJ = MN (CPCTC)

KJ = MN = $\sqrt{5^{2}\;-\;4^{2}}$ (By Pythagoras theorem)

KJ = MN = $\sqrt{25 \;-\; 16}

KJ = MN = $\sqrt{9}$

KJ = MN = 3 units

4. In the following figure, prove that △DAB △CBA.

HL theorem example

Solution: 

In △DAB and △CBA, we have

∠ADB = ∠ACB = 90°

AB = AB (common side)

AD = BC = 4 units

By HL Theorem,

△DAB ≅ △CBA

5. In the following figure, prove that △ABC △EDC.

HL theorem problem

Solution: 

CD = BD – BC = 14 – 8 = 6 units

In △ABC, we have

$AC^{2} = AB^{2} + BC^{2}$

$AC^{2} = 6^{2} + 8^{2} = 36 + 64 = 100$

AC = 10 inches

In △ABC and △EDC,

∠ABC = ∠EDC = 90°

AC = CE = 10 units …hypotenuse

AB = CD = 6 units …leg

By HL Theorem,

△ABC ≅ △EDC

Practice Problems on HL Theorem (Hypotenuse Leg Theorem)

Hypotenuse Leg Theorem – Definition, Proof, Examples, FAQs

Attend this quiz & Test your knowledge.

1

Which information is missing to prove △ABC ≅ △DEF by HL theorem?

Hypotenuse Leg Theorem – Definition, Proof, Examples, FAQs
∠BAC = ∠EDF
AB = ED
AC = DF
∠ACB = ∠EFD
CorrectIncorrect
Correct answer is: AC = DF
In the given right triangle, BC = EF are two congruent legs.
If the two hypotenuses are also equal, we can apply the HL theorem of congruence.
Missing information is AC = DF.
2

Which congruence theorem is used for proving △XYZ ≅ △UZY?

Hypotenuse Leg Theorem – Definition, Proof, Examples, FAQs
SAS
HL
AAS
SSS
CorrectIncorrect
Correct answer is: HL
In △XYZ and △UZY
∠XYZ = ∠UZY = 90°
XZ = YU (given)
YZ = ZY (commons side)
By HL theorem,
△XYZ ≅ △UZY
3

Find the values of a and b that make △WXY ≅ △GFH

Hypotenuse Leg Theorem – Definition, Proof, Examples, FAQs
a = 9, b = 5
a = 5, b = 9
a = 4, b = 9
a = 5, b = 4
CorrectIncorrect
Correct answer is: a = 5, b = 9
For △WXY ≅ △GFH, one leg should be congruent.
WX = GF
2b + 20 = b + 25
b = 5
Also, we need to have two congruent hypotenuses.
WY = GH
11a - 27 = 8a
3a = 27
a = 9

Frequently Asked Questions on HL Theorem (Hypotenuse Leg 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.

Hypotenuse² = Base² + Perpendicular².

According to the leg-leg theorem, if the legs of one right triangle are congruent to the legs of another right triangle, then the triangles are said to be congruent.

If the hypotenuse and one leg (H and L) of a right triangle is proportional to the hypotenuse and leg of the other right triangle, the triangles are said to be similar.