ADVANCED GEOMETRY: OUR PROOF TOOLBOX (Units 1 and 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 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 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.
If two angles are congruent, their supplements are congruent.
Triangle Congruency Theorems: SAS, ASA, SSS, AAS, CPCTC
Triangle Similarity Theorems: AA, SSS, SAS
The midline (or midsegment) of a triangle bisects two sides of the triangle and is parallel to the
third side and ½ the length of the third side.
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
2
2
y y
m
x x
2
1
2
1
Parallel lines have the same slope.
Perpendicular lines have slopes that are opposite reciprocals of one another.
ADVANCED GEOMETRY: OUR PROOF TOOLBOX (Unit 3)
Definitions that can be used to justify steps in a proof:
A regular polygon is a polygon for which all sides are congruent and all angles are congruent.
Alternate Interior Angles – a pair of angles formed by two lines and a transversal. The angles
must lie in the interior of the figure, must lie on alternate sides of the transversal, and must have
different vertices.
Alternate Exterior Angles – a pair of angles formed by two lines and a transversal. The angles
must lie in the exterior of the figure, must lie on alternate sides of the transversal, and must have
different vertices.
Corresponding Angles - a pair of angles formed by two lines and a transversal. One angle must
lie in the interior of the figure, and the other must lie in the exterior. The angles must lie on the
same side of the transversal but have different vertices.
A parallelogram is a quadrilateral with opposite sides parallel.
A rectangle is a parallelogram with four right angles.
A rhombus is a parallelogram with four sides of equal length.
A kite is a quadrilateral with two pairs of congruent adjacent sides.
A trapezoid is a quadrilateral with exactly one pair of parallel sides (per our text).
An isosceles trapezoid is a trapezoid in which the nonparallel sides are congruent. Note: The
nonparallel sides are called the legs of the trapezoid. The parallel sides are called the bases of the
trapezoid.
Theorems that can be used as justification for steps in a proof:
// lines  alt. int . s 
// lines  alt. ext. s 
// lines  corr. s 
alt . int . s   // lines
alt . ext. s   // lines
corr . s   // lines
Opposite sides of a parallelogram are congruent.
Opposite angles in a parallelogram are congruent.
Diagonals of a parallelogram bisect each other.
The diagonals of a rectangle are congruent.
The diagonals of a rhombus bisect the angles of the rhombus.
The diagonals of a rhombus intersect at right angles.
We can also use algebra, a given statement, or a marked diagram to justify a step in a proof,
Formulas that we have justified:
Interior angle sum in a polygon with n sides:
S  180(n  2)
Exterior angle sum (one at each vertex) in a polygon with n sides: E  360
ADVANCED GEOMETRY: OUR PROOF TOOLBOX (Unit 4)
Definitions that can be used to justify steps in a proof:
Polygons are similar if all corresponding angles are congruent and if all corresponding sides are
in the same ratio.
A triangle midsegment is a segment joining the midpoints of two sides a triangle.
In a right triangle, the sine of an acute angle is the ratio of the length of the opposite side to the
length of the hypotentuse.
In a right triangle, the cosine of an acute angle is the ratio of the length of the adjacent side to the
length of the hypotentuse.
In a right triangle, the tangent of an acute angle is the ratio of the length of the opposite side to
the length of the adjacent side.
Theorems that can be used as justification for steps in a proof:
Triangle Similarity Theorems: AA, SSS, SAS
The midline (or midsegment) of a triangle bisects two sides of the triangle and is parallel to the
third side and ½ the length of the third side.
ADVANCED GEOMETRY: OUR PROOF TOOLBOX (Unit 5)
Definitions that can be used to justify steps in a proof:
A central angle of a circle is an angle formed between two radii of the circle.
An arc of a circle consists of two points on a circle and all points on the circle in between the two
points.
A chord of a circle is a segment joining any two points on the circle.
A diameter of a circle is a chord that passes through the center of the circle.
A sector of a circle is the region bound by the rays of a central angle of a circle and bound by the
subtended arc (just like a slice of pie).
A tangent line is a line that intersects the circle at exactly one point.
The point at which the tangent line intersects the circle is called the point of tangency.
A point outside of a circle is considered an external point.
Each segment drawn between and external point and a point of tangency is called a tangent
segment.
A circumscribed angle is an angle formed by two rays tangent to a given circle.
Concentric circles are circles that share the same center.
Tangent circles are two circles that are tangent to the same line at the same point. They can be
internally tangent circles or externally tangent circles.
Theorems that can be used as justification for steps in a proof:
If a line (or segment or ray) passes through the center of a circle and is perpendicular to a chord,
then it bisects the chord.
If a line (or segment or ray) passes through the center of a circle and bisects a chord, then it is
perpendicular to that chord.
The perpendicular bisector of a chord passes through the center of the circle.
Chords in a circle equidistant from the center of the circle are congruent.
Congruent chords in a circle are equidistant from the center.
A radius of a circle drawn to a point of tangency is perpendicular to the tangent line.
The distance from an external point of a circle to each point of tangency is equal.
The measure of an inscribed angle in a circle is ½ the measure of the intercepted arc. (We could
also say “1/2 the measure of a central angle intercepting the same arc”).
Inscribed angles of a circle that intercept the same arc are congruent.
If a triangle is inscribed in a circle so that a side of the triangle lies on the diameter of the circle,
then the triangle is a right triangle.
Formulas that we have justified:
C  d
Circumference (two formulas):
C  2r
Area bound by a Circle: A    r 2
Arclength (two formulas):
l

360
 2r when  is in degrees
l  r when  is a radian measure
Sector Area: A 

360
  r2
Angle Measure Conversion Formulas:
# of degrees =
# of radians =
180


180
 (# of radians )
 (# of deg rees )
2
2
2
Circle centered at the point (h,k) with radius r: ( x  h)  ( y  k )  r
Parabola equation where (a,b) is the focus and y = c is the directrix:
( x  a)2  b2  c2  2(b  c) y
ADVANCED GEOMETRY: OUR PROOF TOOLBOX (Unit 6)
Areas:
Circle: A  r 2
Rectangle: A  l  w
Parallelogram: A  b  h
Trapezoid: A 
Triangle: A 
h
b1  b2 
2
1
bh
2
Volumes:
Cylinder:
Prism:
Sphere:
V    r 2h
V  Bh
Cone:
1
V    r 2h
3
Pyramid:
V 
1
Bh
3
4
V    r3
3
Cavalieri’s Principle: If two space figures have the same cross sectional area at each level, then
they have the same volume.
ADVANCED GEOMETRY: OUR PROOF TOOLBOX (Unit 7)
For Mutually Exclusive Events: P(A or B) = P(A) + P(B)
For Overlapping Events: P(A or B) = P(A) + P(B) – P(A and B)
For Independent Events: P(A and B) = P(A) x P(B)
For Dependent Events: P(A and B) = P(A) x P(B|A)
For Conditional Probability: P(B|A) = P(A and B) / P(A)