TRIANGLES
PARTS, 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 triangle
Legs

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 sides
 Base
-the side that is not congruent to
any side of an isosceles triangle.
Different Types of Triangles
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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
EQUILATERAL
ISOSCELES

SCALENE – no sides congruent
SCALENE
Classifying Triangles by Their
Angles

EQUIANGULAR – all angles are congruent

ACUTE – all angles are acute
EQUIANGULAR

RIGHT – one right angle
ACUTE
RIGHT
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OBTUSE – one obtuse angle
OBTUSE
Congruent Triangles
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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
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If the sides are the same then the triangles are
congruent.
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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
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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
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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
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If two triangles are congruent, we often mark
corresponding sides and angles like this:
is congruent to:
Marking of Congruent Triangles
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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
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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:
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If three sides of one triangle are equal to three sides of
another triangle, the triangles are congruent.
2. SAS (side, angle, side)
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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)
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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)
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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)
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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.
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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" !
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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…
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Triangle AED = Equilateral, Equiangular
Triangle ABC = Equilateral, Equiangular
Triangle ACD = Isoceles, Obtuse
Triangle ACE = Scalene, Right
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So how did you do?
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