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Hypotenuse Leg Theorem – Explanation & Examples

In this article, we’ll learn about the hypotenuse leg (HL) theorem. Like, SAS, SSS, ASA, and AAS, it is also one of the congruency postulates of a triangle.

The difference is that the other 4 postulates apply to all triangles. Simultaneously, the Hypotenuse Leg Theorem is true for the right triangles only because, obviously, the hypotenuse is one of the right-angled triangle legs.

What is Hypotenuse Leg Theorem?

The hypotenuse leg theorem is a criterion used to prove whether a given set of right triangles are congruent.

The hypotenuse leg (HL) theorem states that; a given set of triangles are congruent if the corresponding lengths of their hypotenuse and one leg are equal.

Unlike other congruency postulates such as; SSS, SAS, ASA, and AAS, three quantities are tested, with hypotenuse leg (HL) theorem, two sides of a right triangle are only considered.

Illustration:

Proof of Hypotenuse Leg Theorem

In the diagram above, triangles ABC and PQR are right triangles with AB = RQAC = PQ.

By Pythagorean Theorem,

AC2 = AB2 + BC2 and PQ2 = RQ2 + RP2

Since AC = PQ, substitute to get;

AB2 + BC2 = RQ2 + RP2

But, AB = RQ,

By substitution;

RQ2 + BC2 = RQ2 + RP2

Collect like terms to get;

BC2 =RP2

Hence, ABC ≅△ PQR

Example 1

If PR QS, prove that PQR and PRS are congruent

Solution

Triangle PQR and PRS are right triangles because they both have a 90-degree angle at point R.

Given;

  • PQ = PS (Hypotenuse)
  • PR = PR (Common side)
  • Therefore, by Hypotenuse – Leg (HL) theorem, PQR ≅△ PR.

Example 2

If FB = DB, BA = BC, FB AE and DBCE, show that AE = CE.

Solution

By Hypotenuse Leg rule,

  • BA = BC (hypotenuse)
  • FB = DB (equal side)
  • Since, ∆ AFB≅ ∆ BDC, then ∠A = Therefore, AE = CE

Hence proved.

Example 3

Given that ∆ABC is an isosceles triangle and ∠ BAM = MAD. Prove that M is the midpoint of BD.

Solution

Given ∠ BAM = MAD, then line AM is the bisector of ∠ BAD.

  • AB = AD (hypotenuse)
  • AM = AM (common leg)
  • AMB = AMD (right angle)
  • Therefore, BM = MD.

Example 4

Check whether ∆XYZ and ∆STR are congruent.

Solution

  • Both ∆XYZ and ∆STR are right triangles (presence of a 90 – degree angle)
  • XZ = TR (equal hypotenuse).
  • XY = SR (Equal leg)
  • Hence, by Hypotenuse-Leg (HL) theorem, ∆XYZ ≅∆STR.

Example 5

Given: A=C = 90 degrees, AB = BC.  Show that △ABD DBC.

Solution

Given,

  • AB = BC (equal leg)
  • A=C (right angle)
  • BD = DB (common side, hypotenuse)
  • By, by Hypotenuse-Leg (HL) theorem, △ABD DBC

Example 6

Suppose ∠W = Z = 90 degrees and M is the midpoint of WZ and XY. Show that the two triangles WMX and YMZ are congruent.

Solution

  • WMX and △YMZ are right triangles because they both have an angle of 900 (right angles)
  • WM = MZ (leg)
  • XM = MY (Hypotenuse)
  • Therefore, by Hypotenuse-Leg (HL) theorem, △WMX YMZ.

Example 7

Calculate the value of x in the following congruent triangles.

Solution

Given the two triangles are congruent, then;

⇒2x + 2 = 5x – 19

⇒2x – 5x = -19 – 2

⇒ -3x = – 21

x =- 21/-3

x = 7.

Therefore, the value of x = 7

Proof:

⇒ 2x + 2 = 2(7) + 2

⇒14 + 2 = 16

⇒ 5x -19 = 5(7) – 19

⇒ 35 – 19 = 16

Yes, it worked!

Example 8

If A = C = 90 degrees and AB = BC. Find the value of x and y that will make the two triangles ABD and DBC congruent.

Solution

Given,

ABD DBC

Calculate the value of x

⇒ 6x – 7 = 4x + 2

⇒ 6x – 4x = 2 + 7

⇒ 2x = 9

⇒ x = 9/2

x = 4.5

Calculate the value of y.

⇒ 4y + 25 = 7y – 5

⇒ 4y – 7y = – 5 – 25

⇒ -11y = -30

y = 30/11 =2.73

Therefore, △ABD DBC, when x = 4.5 and y = 2.72.

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