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ADVANCED GEOMETRY: OUR PROOF TOOLBOX
(So far – through unit 2)
Definitions that can be used to justify steps in a proof:
Perpendicular lines, rays and segments form right angles.
A segment bisector divides a segment into two congruent segments.
A midpoint divides a segment into two congruent segments.
An angle bisector divides an angle into two congruent angles.
A segment trisector divides a segment into three congruent segments.
An angle trisector divides an angle into three congruent angles.
Two angles whose measures sum to 90 are complementary.
Two angles whose measures sum to 180 are supplementary.
A circle is the set of all points in a plane equidistant from some fixed point (called the center)
Congruent circles are circles with the same radius.
A triangle median is a segment joining a vertex of the triangle with the midpoint on the
opposite side.
A triangle altitude is a perpendicular segment joining a vertex of the triangle to a point on the
opposite side (though the side may need to be extended).
A line, ray or segment that is perpendicular to another segment and passes
midpoint of that segment is a perpendicular bisector to that segment.
through the
A triangle centroid is the intersection point of the three medians of a triangle (and the balance
point of the triangle.)
A triangle orthocenter is the intersection point of the three altitudes of a triangle.
A triangle circumcenter is the intersection point of the three perpendicular bisectors of a
triangle.
A triangle incenter is the intersection point of the three angle bisectors of a triangle.
Theorems that can be used as justification for steps in a proof:
Vertical Angles are congruent.
Two angles that are supplementary to the same angle are congruent.
(Or, supplements of congruent angles are congruent)
Two angles that are complementary to the same angle are congruent.
(Or, complements of congruent angles are congruent)
Two angles that a supplementary and congruent must be right angles.
Any point that lies on a perpendicular bisector of a segment is equidistant from the endpoints of
that segment.
Any point that lies on an angle bisector is equidistant to each side of the angle.
In any isosceles triangle, base angles are congruent.
If base angles of a triangle are congruent, the triangle is isosceles.
All radii of a given circle (or congruent circles) are congruent.
SAS, ASA, SSS, AAS, CPCTC
Transitive Property (The equivalent of “If a = b and b = c, then a = c)
We can also use algebra, a given statement, or a marked diagram to justify a step in a proof,
Formulas that we have justified:
Distance between two points:
Midpoint between two points:
x  x    y  y 
d
2
2
1
2
2
1
x x ,y  y 


2 
 2
1
2
1
2
Pythagorean Theorem: In any right triangle with legs of length a and b and with hypotenuse of
length c, we can conclude that
Slope (steepness) of a line:
a b  c .
2
m
2
2
y y
x x
2
1
2
1
Parallel lines have the same slope. Perpendicular lines have slopes that are opposite reciprocals.