When two legs of the one right triangle is congruent to the other triangle This was proven by what theorem in triangle congruence?

You've accepted several postulates in this section. That's enough faith for a while. It's time for your first theorem, which will come in handy when trying to establish the congruence of two triangles.

• Theorem 12.2: The AAS Theorem. If two angles and a nonincluded side of one triangle are congruent to two angles and a nonincluded side of a second triangle, then the triangles are congruent.

Figure 12.7 will help you visualize the situation. In the following formal proof, you will relate two angles and a nonincluded side of AB to two angles and a nonincluded side of RST.

Figure 12.7Two angles and a nonincluded side of ABC are congruent to two angles and a nonincluded side of RST.

• Given: Two triangles, ABC and RST, with A ~= R , C ~= T , and ¯BC ~= ¯ST.
• Prove: ABC ~= RST.
• Proof: You need a game plan. If only you knew about two angles and the included side! Then you would be able to use the ASA Postulate to conclude that ABC ~= RST. But wait a minute! Because the measures of the interiorangles of a triangle add up to 180º, and you know two of the angles in are congruent to two of the angles in RST, you can show that the third angle of ABC is congruent to the third angle in RST. Then you'll have two angles and the included side of ABC congruent to two angles and the included side of RST, and you're home free.

	Statements	Reasons
1.	ABC and RST with A ~= R , C ~= T , and ¯BC ~= ¯ST.	Given
2.	mA = mR and mC = mT	Definition of ~=
3.	mA + mB + mC = 180º and mR + mS + mT = 180º	Theorem 11.1
4.	mA + mB + mC = mR + mS + mT	Substitution (step 3)
5.	mA + mB + mC = mA + mS + mC	Substitution (steps 2 and 4)
6.	mB = mS	Algebra
7.	B ~= S	Definition of ~=
8.	ABC ~= RST	ASA Postulate

The HL Theorem for Right Triangles

Whenever you are given a right triangle, you have lots of tools to use to pick out important information. For example, not only do you know that one of the angles of the triangle is a right angle, but you know that the other two angles must be acute angles. You also have the Pythagorean Theorem that you can apply at will. Finally, you know that the two legs of the triangle are perpendicular to each other. You've made use of the perpendicularity of the legs in the last two proofs you wrote on your own. Now it's time to make use of the Pythagorean Theorem.

• Theorem 12.3: The HL Theorem for Right Triangles. If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the triangles are congruent.

There are several ways to prove this problem, but none of them involve using an SSA Theorem. Your plate is so full with initialized theorems that you're out of room. Not to mention the fact that a SSA relationship between two triangles is not enough to guarantee that they are congruent. If you use the Pythagorean Theorem, you can show that the other legs of the right triangles must also be congruent. Then it's just a matter of using the SSS Postulate.

Figure 12.8 illustrates this situation. You have two right triangles, ABC and RST.

Figure 12.8The hypotenuse and a leg of ABC are congruent to the hypotenuse and a leg of RST.

• Given: ABC and RST are right triangles with ¯AB ~= ¯RS and ¯BC ~= ¯ST.
• Prove: ABC ~= RST.
• Proof: You already have a game plan, so all that's left is to execute it.

	Statements	Reasons
1.	ABC and RST are right triangles with ¯AB ~= ¯RS and ¯~= ¯ST.	Given
2.	AB = RS and BC = ST	Definition of ~=
3.	(AC)2 + (BC)2 and (RT)2 + (ST)2 = (RS)2	The Pythagorean Theorem
4.	(AC)2 + (BC)2 = (RT)2 + (ST)2	Substitution (steps 2 and 3)
5.	(AC)2 + (ST)2 = (RT)2 + (ST)2	Substitution (steps 2 and 4)
6.	(AC)2 = (RT)2	Algebra
7.	AC = RT	Algebra
8.	¯AC ~= ¯RT	Definition of ~=
9.	ABC ~= RST	SSS Postulate

SSS, SAS, ASA, and AAS are valid methods of proving triangles congruent, but SSA and AAA are not valid methods and cannot be used. In Figure 12.9, the two triangles are marked to show SSA, yet the two triangles are not congruent. Figure 12.10 shows two triangles marked AAA, but these two triangles are also not congruent.

Figure 12.9These two triangles are not congruent, even though two corresponding sides and an angle are congruent. The two congruent sides do not include the congruent angle!

Figure 12.10These two triangles are not congruent, even though all three corresponding angles are congruent.

Excerpted from The Complete Idiot's Guide to Geometry © 2004 by Denise Szecsei, Ph.D.. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.

To order this book direct from the publisher, visit the Penguin USA website or call 1-800-253-6476. You can also purchase this book at Amazon.com and Barnes & Noble.

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.

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.

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.

• Right Angled Triangle Constructions(RHS)
• Right Angled Triangle

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)

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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.