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Home » Triangles » Congruent Triangles » Congruent Right Triangles

Congruent Right Triangles

Last updated: Feb 6, 2021 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

What are congruent right triangles? We can use the properties of special triangles (like isosceles triangles or right triangles) to help us identify corresponding parts that are equal. Let's look at an example:

Problem

Two right triangles, ΔABC and ΔDEF have an equal hypotenuse and equal leg. Prove that the triangles are congruent.

Congruent Right Triangles

Strategy

Looking at the two triangles, we see we have two equal sides - the legs AB and DE, and the hypotenuses AC and DF - this was given in the problem statement.

We also have one pair of congruent angles- the right angles ∠ABC and ∠DEF, as both triangles are right triangles.

Unfortunately, we can't use the Side-Angle-Side postulate, because the congruent angle is not between the two sides.

It is tempting to try and find another pair of angles, but we simply don't know anything about the other two angles. BUT we do know something else about right triangles, which is the relationship between their legs and the hypotenuse- the Pythagorean theorem: a² + b² = c².

Here we already have 2 of the sides being equal, so we can algebraically show that the third side must also be equal and prove that the triangles are congruent using the Side-Side-Side postulate.

Proof

(1) AB = DE        //given
(2) AC = DF        //given
(3) m∠ABC = m∠DEF=90°          // Definition of right triangle
(4) AB² + BC² = AC²           //Pythagorean theorem
(5) DE² + EF² = DF²           // Pythagorean theorem
(6) AB² + EF² = AC²           //substitution using (1), (2) & (5)
(7) BC = EF                        //(4) & (6)
(8) ΔABC ≅ΔDEF              // Side-Side-Side postulate

You will sometimes see this referred to as yet another method for showing triangle congruency, applicable only to right triangles, called 'Hypotenuse-Leg' (or HL, for short) - if two right triangles have the same hypotenuse and one leg, they are congruent.

« Isosceles Triangles: the Height to the Base Bisects the Apex Angle
Diagonals of Rectangles are of Equal Length »

About the Author

Ido Sarig is a high-tech executive with a BSc degree in Computer Engineering. His goal is to help you develop a better way to approach and solve geometry problems. You can contact him at [email protected]

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About

Welcome to Geometry Help! I'm Ido Sarig, a high-tech executive with a BSc degree in Computer Engineering and an MBA degree in Management of Technology. I'm here to tell you that geometry doesn't have to be so hard! My goal with this website is to help you develop a better way to approach and solve geometry problems, even if spatial awareness is not your strongest quality. Read More…

Geometry Topics

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