AI Tutor
Triangles have six individual parts — three sides and three angles — but you never need to measure all six to know two triangles are identical copies.
Around 300 BCE, the Greek mathematician Euclid demonstrated in his Elements that matching just three specific corresponding pieces locks the entire triangle in place.
How do you know whether an angle is 'included' between two sides, and why does its position completely change your proof?

Included vs. Non-Included Parts

An included angle is the corner directly wedged between two known sides, like the hinge of an open laptop.
An included side is the specific segment connecting two known angle corners directly together.
Because the angle in SAS is included, changing its opening would force the opposite side to stretch or shrink, making the triangle rigid and unique.
What happens when triangles have right angles? Do they follow an exclusive shortcut?

The Hypotenuse-Leg (HL) Shortcut

In a right triangle, the side opposite the 90∘90^\circ angle is the hypotenuse, while the other two shorter sides are called legs.
By the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2), knowing hypotenuse cc and leg aa automatically determines the missing leg as b=c2−a2b = \sqrt{c^2 - a^2}, which is why HL works even though it resembles the non-working SSA pattern.