1. Classification of Triangles

Today, we’re diving into the classification of triangles! Let’s start by discussing how we can categorize triangles based on the lengths of their sides. Can anyone tell me what a scalene triangle is?

Is it a triangle where all three sides have different lengths?

Exactly, great job! A scalene triangle has all sides unequal. Now, who can tell me about an isosceles triangle?

That’s a triangle with two sides that are equal.

Correct! And what about an equilateral triangle?

All three sides are equal!

Well done! Remember, you can use the acronym 'SEE' to help remember this: S for scalene, E for equilateral, and E for isosceles.

That’s a helpful way to memorize them!

Exactly! Now, let’s summarize what we learned about triangles classified by sides: scalene has all sides different, isosceles has two equal, and equilateral has all equal.

Now, let’s shift gears and discuss how triangles can be classified based on their angles. Who remembers how we classify triangles by angles?

I think there are acute, right, and obtuse triangles?

That’s right! An acute triangle has all angles less than 90°.

What about a right triangle?

Good question! A right triangle has one angle that is exactly 90°. And what can you tell me about an obtuse triangle?

It has one angle that is greater than 90°.

Fantastic! You all are doing great. To remember these properties, think 'A for acute, R for right, and O for obtuse.' How about we practice these in different triangles?

That sounds like fun! Let's do it!

Awesome! To summarize, we have acute (all angles < 90°), right (one angle = 90°), and obtuse (one angle > 90°).

Let’s now discuss the triangle inequality theorem. Can someone explain what it says?

It says that the sum of any two sides of a triangle must be greater than the third side.

Excellent! This is crucial because it lets us determine whether three lengths can actually form a triangle. Can anyone recall a situation when this theorem might fail?

If I have sides of lengths 3, 4, and 7, they wouldn’t form a triangle, right?

Precisely! Because 3 + 4 equals 7, you get a degenerate triangle — which doesn’t enclose space like a typical triangle.A degenerate triangle is a triangle where the three vertices are collinear, meaning they all lie on a single straight line. This results in the area of the triangle being zero. As you stated, it is a triangle in which the sum of the lengths of two sides equals the length of the third side. Always remember to check that.