Download CHAPTER 4

CHAPTER 4
Congruent
Triangles
•Congruent –
when two
geometric
figures have the
same size and
shape
SECTION
4-1
Congruent
Figures
•Congruent
Segments –
segments with the
same length
•Congruent Angles –
angles with the
same measure
•Congruent Triangles –
•Congruent Polygons –
two triangles whose
vertices can be paired in
such a way so that
corresponding parts
(angles and sides) of the
triangles are congruent.
two polygons whose
vertices can be paired in
such a way so that
corresponding parts
(angles and sides) of the
polygons are congruent.
1
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
•Interior angles – angles
determined by the sides of
a triangle
•Exterior angle – an angle
that is both adjacent and
supplementary to an
interior angle
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)
•If three sides of one
triangle are
congruent to three
sides of another
triangle, then the
triangles are
congruent (SSS)
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)
2
A Way to Prove Two Segments
or Two Angles Congruent
SECTION
4-3
Using Congruent
Triangles
SECTION
4-4
The Isosceles
Triangle Theorems
•Angles at the
base are called
base angles and
the third angle
is the vertex
angle
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 ≅
•Isosceles
Triangle – is a
triangle with
two legs of
equal length and
a third side
called the base
and
THEOREM 4-1
If two sides of a triangle
are congruent, then the
angles opposite those
sides are congruent
3
Corollary 1
An equilateral triangle is
also equiangular.
Corollary 3
The bisector of the
vertex angle of an
isosceles triangle is
perpendicular to the
base at its midpoint .
Corollary
Corollary 2
An equilateral triangle
has three 60° angles.
THEOREM 4-2
If two angles of a
triangle are congruent,
then the sides opposite
those angles are
congruent.
SECTION
4-5
An equiangular triangle
is also equilateral.
Other Methods of Proving
Triangles Congruent
4
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)
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)
SECTION
4-6
Using More than One Pair
of Congruent Triangles
•Hypotenuse – is
the side opposite
the right angle of
a right triangle.
•Legs – the other
two sides
Ways to Prove Two
Triangles Congruent
•All triangles – SSS,
SAS, ASA, AAS
•Right Triangle - HL
SECTION
4-7
Medians, Altitudes, and
Perpendicular Bisectors
5
•Median – is the
segment with
endpoints that are a
vertex of the
triangle and the
midpoint of the
opposite side
•Perpendicular
bisector – is a line,
ray, or segment that
is perpendicular to a
segment at its
midpoint
•Altitude – the
perpendicular
segment from a
vertex to the
line containing
the opposite
side
•Distance from a point
to a line – is the
length of the
perpendicular
segment from the
point to the line or
plane
THEOREM 4-5
THEOREM 4-6
•If a point lies on the
perpendicular
bisector of a
segment, then the
point is equidistant
from the endpoints
of the segment
•If a point is
equidistant from the
endpoints of a
segment, then the
point lies on the
perpendicular bisector
of a segment.
6
THEOREM 4-7
THEOREM 4-8
•If a point lies on the
bisector of an angle,
then the point is
equidistant from the
sides of the angle.
•If a point is
equidistant from the
sides of an angle,
then the point lies
on the bisector of
the angle.
END
7