Transcript vertex of the triangle

Math 2 Geometry
Based on Elementary Geometry, 3rd ed, by Alexander & Koeberlein
3.3
Isosceles Triangles
Isosceles Triangle
• The two congruent sides are called legs
Isosceles Triangle
• The two congruent sides are called legs
• The third side is the base
Isosceles Triangle
• The two congruent sides are called legs
• The third side is the base
• The point at which the two legs meet is the
vertex of the triangle.
Isosceles Triangle
• The two congruent sides are called legs
• The third side is the base
• The point at which the two legs meet is the
vertex of the triangle.
• The angle formed by the legs is the vertex
angle (opposite the base)
Isosceles Triangle
• The two congruent sides are called legs
• The third side is the base
• The point at which the two legs meet is the
vertex of the triangle.
• The angle formed by the legs is the vertex
angle (opposite the base)
• The angles adjacent to the base are called
base angles.
Vertex
Leg
Leg
Base
Vertex angle
Base angle
Base angle
Vertex
Vertex angle
Leg
Leg
Base angle
Base angle
Base
Informal Definitions
Each angle of a triangle has a unique
angle bisector indicated by a ray or
segment from the angle’s vertex
Informal Definitions
The median is a segment that joins one
angle of a triangle to the midpoint of the
opposite side
Informal Definitions
An altitude is a line segment drawn from a
vertex of a triangle to the opposite side so
that the segment is perpendicular to the
opposite side.
Informal Definitions
The perpendicular bisector of a side of a
triangle is a line perpendicular to the side
that intersects at the midpoint of the side.
Theorem 3.3.1
Corresponding altitudes of congruent
triangles are congruent
Theorem 3.3.2
The bisector of the vertex angel of an
isosceles triangle separates the triangle
into two congruent triangles.
Theorem 3.3.3
If two sides of a triangle are congruent,
then the angles opposite these sides are
also congruent.
Theorem 3.3.3
If two sides of a triangle are congruent,
then the angles opposite these sides are
also congruent.
Alternatively: “The base angles of an
isosceles triangle are congruent.”
Theorem 3.3.4
If two angles of a triangle are congruent,
then the sides opposite these angles are
congruent.
Theorem 3.3.4
If two angles of a triangle are congruent,
then the sides opposite these angles are
congruent.
Note: This is the converse of Theorem 3.3.3:
“If two sides of a triangle are congruent,
then the angles opposite these sides are
also congruent.”
Informal Definitions
• If all three sides of a triangle are
congruent, the triangle is equilateral.
• If all three angles of a triangle are
congruent, the triangle is equiangular.
Corollaries 3.3.5 and 3.3.6
• Corollary 3.3.5: An equilateral triangle is
also equiangular.
• Corollary 3.3.6: An equiangular triangle is
also equilateral.
Definition
The perimeter of a triangle is the sum of
the lengths of its sides. Thus if a, b, and c
are the lengths of the three sides, then the
perimeter P is given by
P=a+b+c
a
c
b
Properties of Scalene Triangles
Sides:
No two are .
Angles:
Sum of s is 180
Properties of Isosceles Triangles
Sides:
Exactly two are 
Angles:
Sum of s is 180
Two s 
Properties of
Equilateral (or Equiangular) Triangles
Sides:
All three are 
Angles:
Sum of s is 180
Three  s
All s measure 60
Properties of Acute Triangles
Sides:
Possibly two or three  sides
Angles:
All s are acute
Sum of s is 180
Possibly two or three  s
Properties of Right Triangles
Sides:
Possibly two  sides
a b c
2
Angles:
2
2
One right 
Sum of s is 180
Possibly two  45 s
Acute s are complementary
Properties of Obtuse Triangles
Sides:
Possibly two  sides
Angles:
One obtuse 
Sum of s is 180
Possibly two acute  s