In a right-angled triangle, the hypotenuse is the longest next which is constantly opposite to the appropriate angle. The hypotenuse foot theorem states that two right triangles room congruent if the hypotenuse and one foot of one right triangle are congruent come the other right triangle"s hypotenuse and leg side. In order to prove any kind of two appropriate triangles congruent, we use the HL (Hypotenuse Leg) to organize or the RHS (Right angle-Hypotenuse-Side) congruence rule. Let united state learn an ext about the hypotenuse leg theorem in this page.

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1.Hypotenuse leg Theorem
2.Hypotenuse foot Theorem Proof
3.Solved examples on Hypotenuse leg Theorem
4. Practice inquiries on Hypotenuse foot Theorem
5. FAQs on Hypotenuse foot Theorem

Hypotenuse foot Theorem


According come the hypotenuse foot theorem, if the hypotenuse and one leg of one right triangle are congruent come the other right triangle"s hypotenuse and also leg side, climate the two triangles are congruent. In various other words, a given collection of appropriate triangles space congruent if the matching lengths of their hypotenuse and also one leg are equal. The hypotenuse foot theorem is a criterion the is provided to prove the congruence that triangles. In the other congruency postulates like, next Side next (SSS), side Angle side (SAS), Angle side Angle (ASA), and Angle Angle next (AAS), three criteria room tested, whereas, in the hypotenuse leg (HL) theorem, only the hypotenuse and also one leg room considered. Observe the following number which mirrors a right-angled triangle v two perpendicular legs and a hypotenuse.

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Hypotenuse leg Theorem Proof


The proof of the hypotenuse foot theorem reflects how a given set of appropriate triangles space congruent if the matching lengths of your hypotenuse and also one leg are equal. Observe the adhering to isosceles triangle abc in which side ab = AC and ad is perpendicular come BC.

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Given: Here, alphabet is an isosceles triangle, AB = AC, and advertisement 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 this triangles, and we know they room equal to every other. AD = AD because they are typical in both the triangles.So, abdominal = AC and advertisement is common.Therefore, a hypotenuse and also a leg pair in two ideal triangles, space satisfying the definition of the HL theorem.We understand that angle B and also C are equal (Isosceles Triangle Property).We likewise know the the angles BAD and CAD are equal.(AD bisects BC, which provides BD equal come CD).Therefore, △ADB ≅ △ADCHence proved.

Important Notes

The Pythagorean theorem says that in a right-angled triangle, the square the the hypotenuse (longest side) is equal to the sum of the squares of the various other two sides (base and also perpendicular). This is represented as: Hypotenuse² = Base² + Perpendicular².According to the HL Congruence rule, the hypotenuse and also one leg are the aspects that are provided to check the congruence that triangles.The HL Congruence rule is similar to the SAS (Side-Angle-Side) postulate. The only distinction is the SAS requirements two sides and also the consisted of angle, whereas, in the HL theorem, the recognized angle is the ideal angle, which is not the consisted of angle in between the hypotenuse and also the leg.

Related posts on Hypotenuse foot Theorem

Check the end the complying with pages pertained to the hypotenuse foot theorem.


Important Topics

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

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Solution:

Following the HL theorem, in △ABC and △PQR: BC = QR (congruent hypotenuse)Thus, y = 13AC = PQ (congruent legs)Thus, x = 5.Therefore, x = 13, y = 5.


Example 2. Fred wondered if the Hypotenuse leg Theorem deserve to be verified using the Pythagorean theorem. Can you discover out?

Solution:

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In the number given above, triangle ABC and also XYZ are right triangles with ab = YZ, AC = XZ.

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By Pythagorean Theorem,

(AC)² = (AB)² + (BC)² and (XZ)² = (XY)² + (YZ)²Since AC = XZ, we have the right to write that: (AB)² + (BC)² = (XY)² + (YZ)²---> (Equation 1)It is provided that abdominal = YZ,Substituting abdominal with YZ in Equation 1:(YZ)² + (BC)² = (XY)² + (YZ)²Solving the equation: we gain (BC)² = (XY)². This method side BC = XY. Hence, △ABC ≅ △XYZ. Thus, v the assist of the Pythagorean theorem, the Hypotenuse leg theorem was proved, which says that if the hypotenuse and one foot of one right triangle are congruent come the other right triangle"s hypotenuse and also leg side, climate the 2 triangles are congruent.


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

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Solution:

It is offered that △PSR and also △PQR are right-angled triangles.PS = QR (equal legs, given)PR = PR (equal and also common hypotenuse)Hence, △PSR ≅ △PQR (by HL rule)