In mathematics, we have geometry as a major branch.
Each major branch has its subbranches.
Geometry has triangles as a subbranch.
In triangles, you must have studied about rightangled triangles.
In a rightangled triangle, the hypotenuse is the longest side and it's always opposite the right angle.
In order to prove the two right triangles congruent, we apply HL or RHS congruence rule.
In this minilesson, you will learn the hypotenuse leg theorem, hypotenuse leg theoremproof, Pythagorean theorem, and hypotenuse theorem.
Lesson Plan
What Is Hypotenuse Leg Theorem?
The hypotenuse leg theorem states that two 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.
For a given set of triangles, they are congruent if the corresponding lengths of their hypotenuse and one leg are equal.
The hypotenuse leg theorem is a criterion used to prove whether a given set of right triangles are congruent.
In congruency postulates, SSS, SAS, ASA, and AAS, three quantities are tested, whereas, in hypotenuse leg (HL) theorem, hypotenuse, and one leg are only considered, that too in case of a right triangle.
Hypotenuse Leg Theorem Proof
Given: Here, ABC is an isosceles triangle, AB = AC.
AD is an altitude line.
Proof:
AD, being an altitude line is perpendicular to BC and forms ADB and ADC as rightangled triangles.
AB and AC are hypotenuse of these triangles, and we know they are equal to each other.
Also, AD = AD because they're the same line.
So AB and AD are equal to AC and AD.
That's a hypotenuse and a leg pair in two right triangles, satisfying the definition of the HL theorem.
Well, we know 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).
\(\Delta ADB \cong \Delta ADC\)
Hence proved.
What Is the Application of the HL Theorem?
Note that the hypotenuse and leg are the elements being used to test for congruence.
Just think! We know the hypotenuse and one other side; the third side can be determined by the Pythagorean Theorem. So, can this be considered a version of the SSS case(sidesideside)?
 The triangles are congruent.
 The remaining third sides are equal.
 The other two angles are equal.
 The Pythagorean theorem states that in a rightangled 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:
(\ Hypotenuse^2 = Base^2 + Perpendicular^2 \)  Hypotenuse equation is \(\ a^2 + b^2 = c^2\). Here, a & b are opposite and adjacent sides.

In the case of the HL Congruence rule, the hypotenuse and leg are the elements, used to test for congruence.

This is kind of like the SAS or sideangleside postulate.
But SAS requires you to know two sides and the included angle.
With the HL theorem, you know two sides and an angle, but the angle you know is the right angle, which isn't the included angle between the hypotenuse and a leg.
Solved Examples
Example 1 
For what values of \(x\) and \(y\), \(\Delta ABC \cong \Delta PQR\)?
Solution
Following the HL theorem,
In \(\Delta ABC\) and \(\Delta PQR\),
BC = QR (congruent hypotenuse)
Thus, \(y\) = \(13\)
AC = PQ (congruent legs)
Thus,\(x\) = \(5\).
\(\therefore\) \(x\) = \(13\), \(y\) =\(5\) 
Example 2 
Fred wondered if Hypotenuse Leg Theorem can be proved using the Pythagorean theorem.
Let's check!
Solution
In the diagram above, triangles ABC and XYZ are right triangles with AB = XY, AC = XZ.
By Pythagorean Theorem,
\(\A C^{2}=A B^{2}+B C^{2} \text { and } X Z^{2}=X Y^{2}+RY Z^{2}\\
&\text { since, } A C=PX Z, \text { substitute to get; }\\
&A B^{2}+B C^{2}=X Y^{2}+Y Z^{2}
\end{aligned}
But, AB = XY,
By substitution,
\begin{aligned}
&Y Z^{2}+B C^{2}=Y Z^{2}+ X Y^{2}\\
&\text { Collect like terms to get; }\\
&B C^{2}=Y Z^{2}\\
&\text { Hence, } \triangle A B C \cong \Delta X Y Z
\end{aligned}
Hence proved. 
Example 3 
For the given figure, prove that \(\Delta PSR \cong \Delta PQR\).
Determine the test of congruence.
Solution
Both are rightangled triangles.
Also, \(\Delta PSR\) and \(\Delta PQR\),
PS = QR (equal legs)
PR = PR (equal hypotenuse)
\(\therefore\) \(\Delta PSR \cong \Delta PQR\) (by HL rule)
\(\therefore\) \(\Delta ABC \cong \Delta PQR\) 

As Christmas is approaching, Mr. William decided to decorate the windows for his floor, i.e., the first floor.
Last time, when he washed the windows, he noticed that all the three windows \(12 \: \text{feet}\) off the ground.
He used up a ladder which was \(13 \: \text{feet}\) long.
Does this mean he placed the base of the ladder away from the building, with the same distance, each time for the three windows?
Does this follow the HL criterion? Can we use the Pythagorean theorem to find how far he should place the ladder each time, for decorating those same three windows?
Interactive Questions
Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.
Let's Summarize
We hope you enjoyed learning about the hypotenuse leg theorem with the simulations and practice questions. Now you will be able to easily solve problems on hypotenuse leg theoremproof, Pythagorean theorem, hypotenuse theorem.
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Frequently Asked Questions (FAQs)
1. What is the isosceles triangle theorem?
According to the isosceles triangle theorem, the angles opposite to the equal sides of an isosceles triangle are also equal.
2. What is the equilateral triangle theorem?
According to the equilateral triangle theorem, if all three sides of a triangle are equal, then all three angles are equal.
3. What is the Pythagorean theorem?
In a rightangled triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides.
4. What is the SSA theorem?
SSA (sidesideangle) refers to one of the criteria of 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.