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G E O M E T R Y
CHAPTER 4
CONGRUENT
TRIANGLES
Notes & Study Guide
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TABLE OF CONTENTS
TRIANGLES AND ANGLES ................................................................................ 3
CONGRUENCE AND TRIANGLES ..................................................................... 7
PROVING TRIANGLES CONGRUENT (SSS/SAS) ............................................ 9
PROVING TRIANGLES CONGRUENT (ASA/AAS)............................................ 9
ISOSCELES, EQUILATERAL & RIGHT TRIANGLES ...................................... 12
EXTRA HOMEWORK EXAMPLES ................................................................... 14
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TRIANGLES AND ANGLES
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INTRODUCTION
In Chapter 4 we work with our first triangle chapter. We will first establish the types of
triangles and their vocabulary. Then we will study the concept of congruence and how it
is applied to triangles.
When we classify (categorize) triangles we do it by looking at the triangles’ sides
and/or its’ angles.
 CLASSIFYING TRIANGLES BY SIDES
To classify a
triangle by its
sides, find out
how many of
the sides are
congruent.
Equilateral  all 3 sides are congruent
Isosceles  at least 2 sides are congruent
Scalene  no sides are congruent
All triangles get
classified both ways
at the same time!
 CLASSIFYING TRIANGLES BY ANGLES
To classify a triangle
by its angles, check
the sizes of the
angles and whether
they are acute, right
or obtuse.
Acute triangle  all 3 angles are acute (< 90°)
Right triangle  must have 1 right angle (= 90°)
Obtuse triangle  must have 1 obtuse angle (> 90°)
Equiangular triangle  type of acute triangle where all 3 angles are equal
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TRIANGLES AND ANGLES
 MORE IMPORTANT TERMS
The following terms are important when discussing triangles and we will use
these throughout the rest of the course…
Vertex  Each of the three “corners” of the triangle
Side  Each of the three segments of the triangle
Adjacent Sides  any two sides of a triangle that
share a vertex (they touch that vertex)
Opposite Side  the side that is directly across
from a particular angle in a triangle (does not touch
that vertex)
Shown here: CA and BA are adjacent to angle A while CB is opposite of A.
RIGHT TRIANGLES
Hypotenuse – the longest side of a right triangle
(it is ALWAYS across from the right angle!)
Legs – the sides that form the right angle in a right
triangle
ISOSCELES TRIANGLES
Legs – the two congruent sides of an isosceles triangle
Base – the third (non-congruent) side of an isosceles
triangle
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TRIANGLES AND ANGLES
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 MAIN TRIANGLE THEOREMS
As we study triangles, we will make use of a couple of key principles. These
theorems will be used repeatedly, so it’s important to know them well. But first…
Interior angles – angles on the inside Exterior angles – angles on the outside
THEOREM 4.1 (Triangle Sum Theorem)
The sum of the measures of the interior angles of a
triangle is ALWAYS 180°.
Shown: angle A + angle B + angle C = 180
THEOREM 4.2 (Exterior Angle Theorem)
The measure of an exterior angle of a triangle is equal
to the sum of the two interior angles across from it.
Shown: angle 1 = angle A + angle B
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TRIANGLES AND ANGLES
 HOMEWORK EXAMPLES
Classify each triangle.
••• Sides of 2cm, 3cm, 4cm
••• Angles of 20°, 145°, 15°
Find the measures of the numbered angles.
Find the measure of the third angle of a triangle given the first two.
••• Angle 1 = 56° Angle 2 = 42°
Angle 3 = ____
••• Angle 1 = 113° Angle 2 = 44°
Angle 3 = ____
Find the value of x in each figure.
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CONGRUENCE AND TRIANGLES
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 CONGRUENCE
When two figures have the same size AND the same shape,
they are called congruent.
To be the same size…all the sides have to match up
To be the same shape…all the angles have to match up
Each pair of sides or angles that match up to each other are called
corresponding sides or angles.
Example: the tick marks show which parts match
up (correspond) to each other…
Sides: AB = PQ, BC = QR, CA = RP
Angles: A = P, B = Q, C = R
In order for two triangles to be congruent, a total of 6 things must happen…
 all 3 pairs of sides must match AND
 all 3 pairs of angles must match
Look for tick marks, angle marks and labeled angles to help you match things up.
 NAMING CONGRUENT FIGURES
When you name figures that are congruent, the order of the letters MUST match
up (correspond) to each other. (even when you read the names, too)
Example: Let’s say the first triangle is DABC .
Since A @ P , B @ Q & C @ R , the name of the
other triangle MUST be named as DPQR .
So, if the 1st one was DBCA , then the second one must be ______.
And, if the 1st one was DCAB , then the second one must be ______.
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CONGRUENCE AND TRIANGLES
 WRITING A CONGRUENCE STATEMENT
A congruence statement is a sentence that just says that one figure is congruent
to another (by name).
Writing one is very easy…
(1) Name the 1st figure (usually many ways to do it)
(2) Name the 2nd figure so that the letters correspond (match
up) to the 1st one.
(3) Put the congruent symbol in between the names.
Example: One possible congruence statement for this is DDEF @ DJKL
Rearrange the name of the first triangle and list some more possible answers.
 HOMEWORK EXAMPLES
Shown at right: DABC @ DTVU
• ÐA @ ____
• BC = _____
• VT @ ____
• DUTV @ ____
Decide if the two figures are
congruent. If they are, write
a congruence statement for
them.
Assume the figures shown are congruent.
Find the values of the variables.
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PROVING TRIANGLES CONGRUENT
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INTRODUCTION
In section 2, we learned that for two triangles to be congruent we had to have 6 things
happen: 3 pairs of congruent sides and 3 pairs of congruent angles.
We’d rather not have to do 6 things to show that two triangles are congruent. So in
sections 3 & 4, we will learn about 4 shortcut Postulates that let us do it in fewer steps.
 TRIANGLE CONGRUENCE POSTUALTES
To prove these are congruent…
…we need to have all of this.
To make it easier, we use Triangle Congruence Postulates to allow us to prove
the same congruence, but with fewer requirements (usually just 3).
POSTULATE 19 •• Side-Side-Side (SSS) Congruence Postulate
If 3 sides of one triangle are congruent to 3 sides of
another triangle, then the two triangles are
congruent.
You don’t need the angles at all. If you’ve got the 3 pairs
of sides, you got it.
POSTULATE 20 •• Side-Angle-Side (SAS) Congruence Postulate
If 2 sides and the included angle of one triangle
are congruent to 2 sides and the included
angle of another triangle, then the triangles are
congruent.
Here you need any 2 sides and the angle that is included (in between) the two sides.
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PROVING TRIANGLES ARE CONGRUENT
POSTULATE 21 •• Angle-Side-Angle (ASA) Congruence Postulate
If 2 angles and the included side of one triangle are
congruent to 2 angles and the included side of another
triangle, then the triangles are congruent.
Here you need any 2 angles and the side that is included (in
between) the two angles.
THEOREM 4.5 •• Angle-Angle-Side (AAS) Congruence Postulate
If 2 angles and the non-included side of one triangle
are congruent to 2 angles and the non-included side of
another triangle, then the triangles are congruent.
You need any 2 angles and the side that comes right after one
of the angles.
THEOREM 4.8 •• Hypotenuse-Leg (HL) Congruence Postulate
If the hypotenuse and leg of a right triangle are
congruent to the hypotenuse and same leg of
another right triangle, then the triangles are
congruent.
Right triangles only! You need the hypotenuse and the
same leg.
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PROVING TRIANGLES ARE CONGRUENT
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HOMEWORK EXAMPLES
Decide whether enough information is given to prove that the triangles are
congruent. If yes, identify which postulate or theorem you would use.
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ISOSCELES, EQUILATERAL & RIGHT TRIANGLES
INTRODUCTION
In this section, we will look at the most common features of isosceles, equilateral and
right triangles.
Knowing these features will help make all work with triangles easier because these
features are almost exclusively used as shortcuts.
 ISOSCELES TRIANGLES
In an isosceles triangle, two sides (the legs) are always
congruent.
Not only are the two legs congruent, the two base angles
(shown in red) are also congruent.
Notice that base angles are directly across (opposite) from the
congruent legs. This pattern is consistent for all triangles and makes a theorem.
THEOREM 4.6 •• Base Angles Theorem
If two sides of a triangle are congruent, then the angles opposite them are also
congruent.
** This theorem is reversible (Converse Theorem 4.7)
 EQUILATERAL TRIANGLES
Equilateral triangles have all sides congruent and the Base
Angle Theorem works on them too.
Since all 3 sides are equal, then the angles opposite those
sides are equal too.
When all the angles of any shape are congruent, we call it equiangular.
If a triangle is equilateral, then it is equiangular.
If a triangle is equiangular, then it is equilateral.
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ISOSCELES, EQUILATERAL & RIGHT TRIANGLES
HOMEWORK EXAMPLES
Find the values of the variable(s) in the triangles.
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EXTRA HOMEWORK EXAMPLES