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Reflexive Property of Congruence
𝐴 𝑓𝑖𝑔𝑢𝑟𝑒 𝑖𝑠 𝑐𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡 𝑡𝑜 𝑖𝑡𝑠𝑒𝑙𝑓.
Definition of Equilateral Quadrilateral
A quadrilateral is equilateral iff it has four congruent sides.
Rhombus
Square
Definition of Base Angle/ Vertex angle
Base Angle: An angle of an isosceles triangle is a base angle iff one of the rays is
the noncongruent side of the triangle and the other ray is one of the congruent
sides of the triangle
Vertex Angle: An angle of an isosceles triangle is a vertex angle iff its vertex is
the endpoint of the two congruent sides (not a base angle!).
Isosceles Theorem
If a triangle is isosceles, then the two base angles of the triangle are congruent.
Converse: If two angles in a triangle are congruent, then the triangle is isosoceles.
Overlapping Angle Theorem
If three angles are adjacent and the outer angles are congruent, then the
overlapping angles are congruent.
If three angles are adjacent and the overlapping angles are congruent, then the
outer angles are congruent.
Recall the overlapping segment theorem:
If four points are collinear and the outer segments are congruent then the
overlapping segments are congruent
If four points are collinear and the overlapping segments are congruent, then
the two outer segments are congruent.
Equilateral Quadrilateral has Opposite Congruent Angles
First define consecutive angles in a polygon:
Two angles are consecutive in a polygon iff the vertices of the two angles are
endpoints of the same side of the polygon.
Now define opposite angles in a quadrilateral:
Two angles are opposite angles in a quadrialateral iff they are not consecutive
angles.
Now we have a new theorem:
Opposite angles in an equilateral quadrilateral are congruent.
Definition of Diagonal
A line segment is a diagonal iff its endpoints are the vertices of opposite angles in
a quadrilateral.
ISO-BAM (part 1)
ISO-BAM has three parts
We have only proven one.
The median of the vertex angle of an isosceles triangle is also the angle
bisector and the altitude.
VA Bisector Theorem
If a ray is the bisector of an angle in a pair of vertical angles, then it’s opposite ray
will bisect the other angle in the vertical angle pair.
Any point on a perpendicular bisector is equidistant
from the end points
Converse: If a point is equidistant from the endpoints of a segment, then it is on
the perpendicular bisector.
Circumcenter is equidistant from the vertices of a
triangle
Circumcenter: A point is the circumcenter of a triangle iff it is the intersection of the
perpendicular bisectors of all three sides.
Uniqueness
The midpoint of a segment is unique
The angle bisector of an angle is unique
The perpendicular bisector of a segment is unique