§3.1 Triangles
The student will learn about:
congruent triangles,
proof of congruency, and
some special triangles.
1
§3.1 Congruent Triangles
The topic of congruent triangles is
perhaps the most used and important in
plane geometry. More theorems are
proven using congruent triangles than
any other method.
2
Triangle Definition
A triangle is the union of three segments
(called its sides), whose end points (called its
vertices) are taken, in pairs, from a set of
three noncollinear points. Thus, if the
vertices of a triangle are A, B, and C, then
its sides are AB , BC and AC , and the
triangle is then the set defined by
AB  BC  AC , denoted ΔABC. The angles
of ΔABC are A  BAC, B  ABC, and
C  ACB.
3
Euclid
Euclid’s idea of congruency involved the act of
placing one triangle precisely on top of
another. This has been called superposition.
4
CONGRUENCY
Definitions
Angles are congruent if they have the same
measure.
Segments are congruent if they have the
same length.
5
Definition
Two triangles are congruent iff the six parts of
one triangle are congruent to the corresponding
six parts of the other triangle.
One concern should be how much of this
information do we really need to know in order
to prove two triangles congruent.
Congruency is an equivalence relation –
reflexive, symmetric, and transitive.
6
Properties of Congruent Triangles
We know that corresponding parts of
congruent triangles are congruent. We
abbreviate this fact as CPCTC and find it quite
useful in proofs.
7
Important Note
When we writeΔ ABC  Δ DEF we are
implying the following:
• A  D
• B  E
• C  F
• AB  DE
• BC  EF
• AC  DF
Order in the statement, Δ ABC  Δ
DEF,
8
We Will Use CPCTE To Establish
Three Types of Conclusions
1. Proving triangles congruent, like
Δ ABC and Δ DEF.
2. Proving corresponding parts of
congruent triangles congruent, like
AB  DE
3. Establishing a further relationship, like
A
B.
9
Some Postulate
Postulate 12. The SAS Postulate
Every SAS correspondence is a congruency.
Postulate 13. The ASA Postulate
Every ASA correspondence is a congruency.
Postulate 12. The SSS Postulate
Every SSS correspondence is a congruency.
10
Marking Drawings
B
A
D
C
AB  CD
A   C
AC  BD
CBD   BCA
AC  BD
11
Suggestions for proofs that
involve congruent triangles:
Mark the figures systematically, using:
A. A square
square in
in the
the opening
opening of
of aa right
right
triangle;
B. The same number of dashes on congruent
sides; And.
C. The same number of arcs on congruent
angles.
D. Use coloring to accomplish the above.
F. If the triangles overlap, draw them
separately.
12
Example Proof
Given: AR and BH bisect
each other at F
Prove: AB  RH
H
A
F
R
B
Statement
1. AR and BH bisect each other.
Reason
Given
2. AF = FR and BF = FH
3. AFB = RFH
4. ∆AFB = ∆RFH
Definition of bisect.
Vertical Angle Theorem
ASA
5. AB = RH
CPCTE
6. QED
13
Definition – Angle Bisector
If D is in the interior of BAC, and BAD is
congruent to DAC then AD bisects BAC, and
AD is called the bisector of BAC.
B
D
A
C
14
Definition – Special Triangles
A triangle with two congruent sides is called
isosceles. The remaining side is the base. The two
angles that include the base are base angles. The
angle opposite the base is the vertex angle.
A triangle whose three sides are congruent is
called equilateral.
A triangle no two of whose sides are congruent is
called scalene.
A triangle is equiangular if all three angles are
congruent.
15
Theorem - Isosceles Triangle
Theorem
The base angles of an Isosceles triangle are
congruent.
Proof is a homework assignment.
16
Theorem – Converse of the
Isosceles Triangle Theorem
If two angles of a triangle are congruent, then the
sides opposite them are congruent.
Proof is a homework assignment.
17
Definition – Right Triangles
A triangle with one right angle is a right triangle.
Because two right triangles automatically have
one angle congruent (the right angle),
congruency of two right triangles reduces to
two cases:
1) HA which is equivalent to ASA since all angles
are known, and
2) HL which is equivalent to SSS since all three
sides are know.
We are assuming knowledge of angle sums and Pythagoras.
18
Assignment: §3.1