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CHAPTER 4
Congruent
Triangles
SECTION
4-1
Congruent
Figures
•Congruent Triangles –
two triangles whose
vertices can be paired in
such a way so that
corresponding parts
(angles and sides) of the
triangles are congruent.
•Congruent Polygons –
two polygons whose
vertices can be paired in
such a way so that
corresponding parts
(angles and sides) of the
polygons are congruent.
SECTION
4-2
Some Ways to Prove
Triangles Congruent
•Included angle –
angle between two
sides of a triangle
•Included side – the
side common to two
angles of a triangle
Postulate 12
•If three sides of one
triangle are
congruent to three
sides of another
triangle, then the
triangles are
congruent (SSS)
Postulate 13
• If two sides and the
included angle of one
triangle are congruent
to two sides and the
included angles of
another triangle, then
the triangles are
congruent (SAS)
Postulate 14
If two angles and the included
side of one triangle are
congruent to two angles and
the included side of another
triangle, then the triangles
are congruent (ASA)
SECTION
4-3
Using Congruent
Triangles
A Way to Prove Two Segments
or Two Angles Congruent
1. Identify two triangles in which
the two segments or angles are
corresponding parts.
2. Prove that the triangles are
congruent
3. State that the two parts are
congruent, use the reason
Corresponding parts of  ∆ are 
SECTION
4-4
The Isosceles
Triangle Theorems
THEOREM 4-1
If two sides of a triangle
are congruent, then the
angles opposite those
sides are congruent
Corollary 1
An equilateral triangle is
also equiangular.
Corollary 2
An equilateral triangle
has three 60° angles.
Corollary 3
The bisector of the
vertex angle of an
isosceles triangle is
perpendicular to the
base at its midpoint .
THEOREM 4-2
If two angles of a
triangle are congruent,
then the sides opposite
those angles are
congruent.
Corollary
An equiangular triangle
is also equilateral.
SECTION
4-5
Other Methods of Proving
Triangles Congruent
THEOREM 4-3
• If two angles and a
non-included side of
one triangle are
congruent to the
corresponding parts of
another triangle, then
the triangles are
congruent (AAS)
•Hypotenuse – is
the side opposite
the right angle of
a right triangle.
•Legs – the other
two sides
THEOREM 4-4
• If the hypotenuse and a
leg of one right triangle
are congruent to the
corresponding parts of
another right triangle,
then the triangles are
congruent. (HL)
Ways to Prove Two
Triangles Congruent
•All triangles – SSS,
SAS, ASA, AAS
•Right Triangle - HL
SECTION
4-6
Using More than One Pair
of Congruent Triangles
SECTION
4-7
Medians, Altitudes, and
Perpendicular Bisectors
•Median – is the
segment with
endpoints that are a
vertex of the
triangle and the
midpoint of the
opposite side
•Altitude – the
perpendicular
segment from a
vertex to the
line containing
the opposite
side
•Perpendicular
bisector – is a line,
ray, or segment that
is perpendicular to a
segment at its
midpoint
THEOREM 4-5
•If a point lies on the
perpendicular
bisector of a
segment, then the
point is equidistant
from the endpoints
of the segment
THEOREM 4-6
•If a point is
equidistant from the
endpoints of a
segment, then the
point lies on the
perpendicular bisector
of a segment.
•Distance from a point
to a line – is the
length of the
perpendicular
segment from the
point to the line or
plane
THEOREM 4-7
•If a point lies on the
bisector of an angle,
then the point is
equidistant from the
sides of the angle.
THEOREM 4-8
•If a point is
equidistant from the
sides of an angle,
then the point lies
on the bisector of
the angle.
END