TRIANGLESPARTS, CLASSIFICATIONS,

ANGLES NAD PROVING CONGRUENCE OF

TRIANGLES.

PARTS OF TRIANGLES

Sides the edges or boundaries of the triangle.

Vertices part where the two sides join.

Adjacent sides two sides that have common vertex

PARTS OF TRIANGLES

In a right triangleLegs

the sides adjacent to the right angle in a right triangle.

Hypotenuse the side opposite the right angle in a right angle.

PARTS OF TRIANGLES

In an isosceles triangle,Legs

-the congruent sidesBase

-the side that is not congruent to any side of an isosceles triangle.

Different Types of Triangles

There are several different types of triangles.

You can classify a triangle by its sides and its angles.

There are THREE different classifications for triangles based on their sides.

There are FOUR different classifications for triangles based on their angles.

Classifying Triangles by Their Sides

EQUILATERAL – 3 congruent sides

ISOSCELES – at least two sides congruent

SCALENE – no sides congruent

EQUILATERAL

ISOSCELES

SCALENE

Classifying Triangles by Their Angles

EQUIANGULAR – all angles are congruent

ACUTE – all angles are acute

RIGHT – one right angle

OBTUSE – one obtuse angle

EQUIANGULAR

ACUTE

RIGHT

OBTUSE

Congruent Triangles

What is "Congruent" ... ?It means that one shape can

become another using Turns, Flips and/or Slides:

ROTATION REFLECTION

TRANSLATION

Congruent Triangles

If two triangles are congruent they will have exactly the same three sides and exactly the same three angles.

The equal sides and angles may not be in the same position (if there is a turn or a flip), but they will be there.

Same Sides of a Triangle

If the sides are the same then the triangles are congruent. For example:

is congruent to and

because they all have exactly the same sides.

Same Sides of a Triangle

If the sides are the same then the triangles are congruent. For example:

is not congruent to

because the two triangles do not have exactly the same sides.

Same Angles of a Triangle

Does this also work with angles? Not always!

Two triangles can have the same angles but be different sizes:

is not congruent to

because, even though all angles match, one is larger than the

other.

Same Angles of a Triangle

Can two triangles of the same angles be congruent?

Yes. They could be congruent if they are the same size

is congruent to

because they are (in this case) the same size

Marking of Congruent Triangles

If two triangles are congruent, we often mark corresponding sides and angles like this:

is congruent to:

Marking of Congruent Triangles

The sides marked with one line are equal in length. Similarly for the sides marked with two lines and three lines.

The angles marked with one arc are equal in size. Similarly for the angles marked with two arcs and three arcs.

How To Find if Triangles are Congruent

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:

is congruent to:  

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:is congruent to:  

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:is congruent to:  

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 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:is congruent to:  

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)

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

and  

HL applies only to right angled-triangles!

5. HL   (hypotenuse, leg) 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:

is congruent to 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.

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

Because the triangles can have the same angles but be different sizes:

For example: is congruent to 

Without knowing at least one side, we can't be sure if two triangles are congruent..

Can You Classify the Different Triangles in the

Picture Below?

Classify the following triangles: AED, ABC, ACD, ACE

The Classifications…

Triangle AED = Equilateral, Equiangular

Triangle ABC = Equilateral, Equiangular

Triangle ACD = Isoceles, Obtuse Triangle ACE = Scalene, Right

So how did you do?