Download Justifications So Far

Name: _________________________
Geometry – Class 823
Justifications You Can Use in Proofs (So far)
Basic Properties:
 Transitive Property (A=B, B=C, then A=C)
 Reflexive Property (A = A)
 Symmetry Property (AB = BA)
 Addition property of equality
 Subtraction property of equality
 Segment Addition Postulate
 Angle Addition Postulate
 There exists a unique parallel line that can be drawn through a point off a line. (Euclid’s parallel
postulate)
Parallel Lines:
 If parallel lines are cut by a transversal, then Alternate Interior Angles are congruent.
 If parallel lines are cut by a transversal, then Alternate Exterior Angles are congruent.
 If parallel lines are cut by a transversal, then Corresponding Angles are congruent.
 If Corresponding Angles are congruent, then the lines are parallel
 If Alternate Interior Angles are congruent, then the lines are parallel
 If Alternate Exterior Angles are congruent, then the lines are parallel
 The midsegment of a triangle is parallel to the opposite side
Angle Properties:
 Vertical Angles are congruent
 Supplemental Angles add up to 180 degrees
 Corresponding Angles add up to 90 degrees
 The interior angles of a triangle add up to 180 degrees
 The exterior angle of a triangle is equal to the sum of the opposite interior triangles
 Supplements of congruent angles are congruent
 Complements of congruent angles are congruent
Triangle Congruence:
 SAS
 ASA
 AAS
 SSS
 HL
 CPCTC (Corresponding Parts of Congruent Triangles are Congruent) – This can be used after
establishing triangles are congruent to show that the corresponding parts are congruent– Not in
order to prove the triangles as a whole are congruent!
Quadrilateral Properties:
 In a parallelogram, opposite sides are congruent
 In a parallelogram, opposite angles are congruent
 In a parallelogram, the diagonals bisect each other
 In a parallelogram, opposite sides are parallel
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In a rhombus, the diagonals intersect at right angles
In a rhombus, all sides are congruent
In a rhombus, the diagonals bisect the angles
In a rectangle, the diagonals are congruent
In a rectangle, all angles are right angles
In a square, all sides are congruent and all angles are right angles
To Prove A Quadrilateral Is A…:
 Parallelogram
o Both pairs of opposite sides congruent
o Both pairs of opposite angles congruent
o Diagonals bisect each other
o Both pairs of opposite sides are parallel
o One pair of sides that is both congruent and parallel
 Rhombus
o Parallelogram + Consecutive sides congruent
o Parallelogram + Diagonals intersect at right angles
o Parallelogram + Diagonals bisect corners
 Rectangle
o Parallelogram + Congruent Diagonals
o Parallelogram + At least one right angle
o Parallelogram + Diagonals bisect corners
 Square
o Parallelogram + Consecutive sides congruent and one right angle
Circles:
 In a circle, a radius perpendicular to a chord bisects the chord and the arc.
 In a circle, a radius that bisects a chord is perpendicular to the chord.
 In a circle, the perpendicular bisector of a chord passes through the center of the circle.
 If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of tangency.
 In a circle, or congruent circles, congruent chords are equidistant from the center.
 The arcs between congruent chords are congruent
 In a circle, the arcs between parallel chords are congruent
 In the same circle, or congruent circles, congruent central angles have congruent chords (and
converse)
 Tangent segments to a circle from the same external point are congruent
 Tangents drawn from the same point are congruent
 In the same circle, or congruent circles, congruent central angles have congruent arcs.
 An angle inscribed in a semi-circle is a right angle.
 In a circle, inscribed angles that intercept the same arc are congruent.
 The opposite angles in a cyclic quadrilateral are supplementary
 In a circle, or congruent circles, congruent central angles have congruent arcs.
Definitions:
 Definition of midpoint (Use it to justify that segments are congruent on each side of the midpoint)
 Definition of angle bisector (Use it to justify that the angles formed by the bisector are congruent)
 Definition of perpendicular line (Use it to justify saying an angle is 90 degrees)