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## Algebra basics

### Course: Algebra basics>Unit 8

Lesson 4: Triangle similarity intro

# Intro to triangle similarity

Sal explains what it means for triangles to be similar, and how this follows from the definition of similarity. Created by Sal Khan.

## Want to join the conversation?

• What do you mean by scale factor? I thought that was only used to measure 3D objects. •   Scale Factor is the multiplier used on each dimension to change one figure into a similar figure.
For example, graph these points: It will create a triangle.
A (2,3)
B (0,1)
C (3,0)
then these will be will create another triangle:
A' (4,6)
B' (0,2)
C' (6,0)
You would then choose two coordinates, such as C and C', remove their zeros and divide them, and that's your scale factor: 2.
Note: A scale factor can be used in 1, 2, even 3 dimensions. I hope that made sense.
• What is "congruent?" • when he says at around that all of their corrisponding angles are the same, does that mean if lets say angl. A is 30 degrees that angle X will also be the same or that they are in the same scale? • What is "K" supposed to mean?
It is around . •  He is calling K the scaling factor. It is just a number that represents how the similar figures sides relate to the original figure.

If K > 1 then the similar figure is scaled up. That means it is bigger. For example, if K = 5 then each side of the similar figure will be 5 times the original figure.

If K < 1 (a fraction), it means that the similar figure is scaled down. For example, if K = 1/4, then each side of the similar figure will be 1/4 the length of the original figure. [This is like scaled versions of cars, planes, trains, etc.]
• Say you have 2 shapes. If one shape can become another using Turns, Flips and/or Slides, then the shapes are Congruent. After any of those transformations (turn, flip or slide), and the shape still has the same size, area, angles and line lengths, then the shape is congruent to the other. Congruent? Why such a funny word that basically means "equal"? Maybe because they are only "equal" when placed on top of each other. Anyway it comes from Latin , "to agree". So the shapes "agree". f you have congruent angles then... They don't have to point in the same direction. They don't have to be on similar sized lines. Just the same angle. Congruent triangles:
Two triangles are congruent if they have: exactly the same three sides and exactly the same three angles. But we don't have to know all three sides and all three angles ...usually three out of the six is enough. There are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS and HL.
1. SSS (side, side, side)

SSS stands for "side, side, side" and means that we have two triangles with all three sides equal.

For example:

If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.

2. SAS (side, angle, side)

SAS stands for "side, angle, side" and means that we have two triangles where we know two sides and the included angle are equal.

For example:

If two sides and the included angle of one triangle are equal to the corresponding sides and angle of another triangle, the triangles are congruent.

3. ASA (angle, side, angle)

ASA stands for "angle, side, angle" and means that we have two triangles where we know two angles and the included side are equal.

For example:

If two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

4. AAS (angle, angle, side)

AAS Triangle

AAS stands for "angle, angle, side" and means that we have two triangles where we know two angles and the non-included side are equal.

For example:

If two angles and the non-included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

5. HL (hypotenuse, leg)

This one applies only to right angled-triangles!

HL stands for "Hypotenuse, Leg" (the longest side of a right-angled triangle is called the "hypotenuse", the other two sides are called "legs")

It means we have two right-angled triangles with

the same length of hypotenuse and
the same length for one of the other two legs.
It doesn't matter which leg since the triangles could be rotated.

For example:

If the hypotenuse and one leg of one right-angled triangle are equal to the corresponding hypotenuse and leg of another right-angled triangle, the two triangles are congruent.

Caution ! Don't Use "AAA" !

AAA means we are given all three angles of a triangle, but no sides.

AAA Triangle

This is not enough information to decide if two triangles are congruent!

Because the triangles can have the same angles but be different sizes:
Without knowing at least one side, we can't be sure if two triangles are congruent. • Are all triangles within another triangle always similar ?
Like a smaller triangle inside a bigger triangle or else smaller triangles made by intersecting lines from originating from the vertices of the bigger triangle ? • Would all these rules work with quadrilaterals as well? if not please tell me which will not. Thanks. • With quadrilaterals you have to check both that the corresponding angles are the same and that the sides are scaled up/down versions of each other for similarity. With triangles having same angles will imply that the sides are scaled versions of each other while at the same time having scaled up/down sides will imply that the corresponding angles have the same measures.   