4-1 Classifying Triangles
 A triangle is a three-sided polygon.
 A triangle is made up of sides, angles, and vertices.
 A triangle can be classified by sides and by angles.
When a triangle is classified by its sides, it can be scalene, isosceles or equilateral.
Label the following triangles as scalene, isosceles, or equilateral.
___________________
All 3 sides are different lengths.
___________________
At least two of the sides are the
same length.
___________________
ALL 3 sides are the same length
When a triangle is classified by its angles, it can be acute, obtuse, right or equiangular.
Label the following triangles as acute, obtuse, right, or equiangular.
All of the angles are
equal.
___________________
All the angles are less
than 90.
___________________
One of the angles is
exactly 90.
___________________
One of the angles is
greater than 90.
Your Turn!
Match the following triangles to their descriptions.
_______1.
a.
Right scalene
_______2.
b.
Acute isosceles
_______3.
c.
Obtuse isosceles
_______4.
d.
Right isosceles
_______5.
e.
Equiangular equilateral
Your Turn, Again!
Classify the following triangles by sides AND by angles. You will have two answers for each!!
___________________ , ___________________
___________________ , ___________________
___________________ , ___________________
___________________ , ___________________
Special Triangles
Right Triangles
Right triangles have two legs and a hypotenuse. The hypotenuse is the always the side that is
across from the right angle. One special thing about a right triangle is that the two angles other than
the right angle are always acute and complementary.
Isosceles Triangles
Isosceles triangles have two legs and a base. The base is the bottom of the triangle. Isosceles
triangles also have two base angles and a vertex angle. The base angles are on either side of
the base and are congruent. The vertex angle is in between the two congruent sides.
Example 1
Triangle ABC is an equilateral triangle. Find x and the measure of each side if AB = 4x - 3 and
BC = 3x + 4.
4-2 Angles of Triangles
 The Angle Sum Theorem states that the sum of the measures of the interior angles of a
triangle is always 180o.
Example 1
Find the measure of <T.
Our Turn!
Solve for each numbered angle.
 The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is
equal to the sum of the measures of the two remote interior angles.
Example 2
Find m<1.
Our Turn!
Solve for each numbered angle.
Your Turn!
Solve for each numbered angle.
4-3 Congruent Triangles
Recall from Chapter 1 that the word congruent means equal, or the same. Similarly, if two triangles
are congruent they will have exactly the same three sides and exactly the same three
angles.
Your Turn!
Which triangle is not congruent to the other three?
 CPCTC states that
C ______________________
P _____________ of
C _________________
T ________________ are
C ________________
Two triangles are congruent ONLY if their corresponding parts are congruent. This means the
corresponding angles must be congruent and the corresponding sides must be congruent as well.
Example 1
Take a look at the example below. Triangles RST and XYZ are congruent.
Name all the congruent angles.
Name all the congruent sides.
 Congruency statements will tell you what angles and sides are equal in triangles.
Example 2
Write a congruency statement for the triangles above.
Your Turn!
Using the congruency statement below, name the congruent angles and sides.
Name all the congruent angles.
Name all the congruent sides.
 Turns, flips, and slides are called congruence transformations. They do not
change the congruence of the triangle.
Slide
Flip
Turn
Your Turn!
Circle the right description for the transformations below.
4-4 and 4-5 Proving Triangles 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.
Ways to Determine Triangle Congruence
There are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS and HL.
1. SSS (_______,_______,_______)
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 (_______,_______,_______)
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 (_______,_______,_______)
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 (_______,_______,_______)
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 (_____________________, _______)
This one applies only to right triangles!
or
HL stands for "Hypotenuse, Leg" (the longest side of the triangle is called the "hypotenuse", the
other two sides are called "legs")
For example:
is congruent to:
If the hypotenuse and one leg of one right triangle are equal to the corresponding hypotenuse and
leg of another right triangle, the two triangles are congruent.
Caution !
Don't Use "AAA" OR the Donkey Theorem!
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:
is not congruent to:
Without knowing at least one side, we can't be sure if two triangles are congruent.
The Donkey Theorem is pretty self-explanatory. DON’T use it!
Your Turn!
Draw two triangles that are congruent by each of the following congruence theorems.
SSS
SAS
If three sides of one triangle are equal to three
sides of another triangle, the triangles are
congruent.
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.
ASA
AAS
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.
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.
HL
If the hypotenuse and one leg of one right triangle
are equal to the corresponding hypotenuse and leg
of another right triangle, the two triangles are
congruent.