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Handout Page 1
How do I prove?
Congruent Angles
1. Corresponding parts of triangles
2. Definition of angle bisector
3. In a , angles opposite sides
4. Perpendicular lines
5. Complements to the same angles
6. Complements to congruent angles
7. Supplements to the same angles
Congruent Segments
1. Corresponding parts of triangles
2. Definition of a segment bisector
3. Definition of midpoint
4. In a sides opposite ’s
Congruent Triangles
1. Side Angle Side
3. Side Side Side
5. Hypotenuse Leg
7. Definition of triangles
Perpendicular Lines
1. Definition of perpendicular lines
2. Congruent adjacent angles
3. Definition of an altitude
Parallel Lines
1. Definition of parallel lines
2. *Alternate interior ’s 
3. *Corresponding ’s 
4. *Interior ’s same side, supplementary
* - Angles formed by a transversal
5. Two lines to a third
Right Angles
1. Definition of right angles
2. Definition of perpendicular lines
3. Definition of a rectangle
8. Supplements to congruent angles
9. Right angles
10. Vertical angles
11. Definition of congruence
12. *Alternate interior angles (||)
13. Opposite angles of a parallelogram
14. *Corresponding angles (||)*Angles
formed by a transversal.
5. Definition of congruence
6. Definition of a median
7. Opposite sides of a parallelogram
8. Diagonals of a rectangle
2. Angle Side Angle
4. Hypotenuse Acute Angle
6. Side Angle Angle
4. Two lines on a plane || to a third
5. Diagonals of a rhombus
6. Two lines || to a third
7. *Alternate exterior ’s 
8. Definition of a parallelogram
9. Definition of a trapezoid
Handout Page 2
Supplementary Angles
1. Definition of supplementary angles
2. Linear pairs
Angle Bisector
1. Point equidistant to sides
2. Diagonals of a rhombus
3. Definition of an angle bisector
Perpendicular Bisector
1. Definition of perpendicular bisector
2. Two points equidistant to endpoints
Parallelogram
1. Definition of a parallelogram
2. Both pairs of opposite sides 
3. One pair of sides both and ||
Similar Triangles
1. Definition of similar triangles
3. AA corollary
5. SAS Similarity theorem
7. Side || to Third Side
3. *Interior ’s on same side * - Angles
formed by a transversal
4. Consecutive angles of a parallelogram
3. Bisector of a vertex angle
2. AAA Theorem
4. Acute angles of right triangles
6. SSS Similarity Theorem
Theorems, Corollaries & Postulates
A postulate is a mathematical statement that is accepted without proof.
A theorem is a mathematical statement that can be proved.
A corollary is a theorem that follows directly from a theorem or accepted statements,
such as definitions.
A conditional statement is a statement in the “if-then” form.
The hypothesis is the “if” part of a conditional statement or what is accepted as a fact
(given).
The conclusion is the “then” part of a conditional statement or what is to be proved
(prove).
Proofs; formal and informal
Postulate • a mathematical statement that is accepted w/o proof
Theorem • a mathematical statement that can be proved
Conditional statement • a statement in the “if-then” form
Hypothesis • the “if” part of a conditional statement; the accepted facts (given)
Conclusion • the “then” part of a conditional statement; what to be proved (prove)
Deductive reasoning • the thought process used to prove theorems
1. You begin with the accepted fact or facts (hypothesis).
2. You then proceed in a step-by-step fashion with statements that are justified by
stating the hypothesis, postulates, theorems, or definitions as reasons.
3. Each statement should lead you to another statement until you reach the last statement
that is the conclusion.
Formal proof • step by step statements with reasons showing a complete sequence of
proof (two forms)
a. Two-column
b. Paragraph
Converse • conditional statement when the “if” and “then” are reversed
Biconditional • “iff”... a conditional statement and its converse stated as one with “if and
only if”
Contradiction • a mathematical statement that is both true and false.
Negation • to state the opposite of the original statement.
Indirect proof • prove something is true by proving that it is not false
1. Assume the negation of the prove.
2. Ask the questions of a two column proof
• Use the assumption and the true given information.
3. Do the proof until you get a contradiction.
• Prove the negation of a given - Contradiction
Handout Page 3
Postulates; Basic
1. Every line contains at least two distinct points.
2. Every plane contains at least three distinct non-collinear points.
3. Space contains at least four distinct non-coplanar points.
4. For any two distinct points, there is exactly one line containing them.
• Two distinct points determine a line.
5. For any three distinct non-collinear points, there is exactly one plane containing them.
• Three distinct non-collinear points determine a plane.
6. If any two distinct points lie in a plane, the line containing these points lies on the
plane.
7. If two distinct planes have one point in common, they have at least two points in
common.
Theorems; Basic
1. If two distinct planes intersect, then their intersection is a line.
2. If two distinct lines intersect, then their intersection is a point.
3. If a line and a plane intersect and the plane does not contain the line, then their
intersection is a point.
4. If a point is not on a line, then the point and the line determine exactly one plane.
5. If two distinct lines intersect, they determine exactly one plane.
Theorems; Points Lines, and Angles
1. A line segment has exactly one midpoint.
2. An angle has exactly one bisector.
3. Through a given point on a line in a plane, there is exactly one line in the plane
perpendicular to the line.
4. For any given segment in a plane, the segment has exactly one perpendicular bisector
in the plane.
1. If two angles form a linear pair, then they are supplementary.
6. If two angles are complements of the same angle, then they are congruent.
7. If two angles are complements of congruent angles, then they are congruent.
8. If two angles are supplements of the same angle, then they are congruent.
9. If two angles are supplements of the congruent angles, then they are congruent.
10. Vertical angles of the same pair are congruent.
11. If one angle of a linear pair is a right angle, then the other is also a right angle.
12. If two intersecting lines form a right angle, then they form four right angles.
• Perpendicular lines form four right angles. • Right angles are congruent.
13. If two intersecting lines form congruent adjacent angles, then the two lines are
perpendicular.
14. If two lines are perpendicular, then they form congruent right angles.
Postulates; Congruent Triangles
1. If two triangles have two sides and the included angle of one triangle congruent
respectively to two sides and the included angle of the other triangle, then the triangles
are congruent. SAS
2. If two triangles have two angles and the included side of one triangle congruent
respectively to two angles and the included side of the other triangle, then the triangles
are congruent. ASA
3. If two triangles have three sides of one triangle congruent respectively to three sides
of the other triangle, then the triangles are congruent. SSS
4. If two right triangles have the hypotenuse and an acute angle of one triangle
congruent respectively to the hypotenuse and an acute angle of the other triangle then
the triangles are congruent. HA
Theorems; Congruent Triangles
1. If two sides of a triangle are congruent, then the angles opposite these sides are
congruent. (Isosceles Triangle).
2. If two angles of a triangle are congruent, then the sides opposite these angles are
congruent.
3. The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector
to the base.
4. 5. The median from the vertex angle of an isosceles triangle is perpendicular to the
base and bisects the vertex angle.
6. In a isosceles triangle, the altitude bisects the base and bisects the vertex angle.
7. If three sides of a triangle are congruent, then the angles opposite these sides are
congruent. • In a triangle, angles opposite congruent sides are congruent.
8. If three angles of a triangle are congruent, then the sides opposite these angles are
congruent. • In a triangle, sides opposite congruent angles are congruent.
9. Any point on the perpendicular bisector of a given segment is equidistant from the
endpoints.
10.In a plane, a line is the perpendicular bisector of a given segment, if and only if, it is
determined by two points equidistant from the endpoints of the segment.
11.If two right triangles have the hypotenuse and one leg of one triangle congruent
respectively to the hypotenuse and the leg of the other triangle, then the triangles are
congruent. (HL)
12. Through a point not on a line, there is exactly one line perpendicular to a given line.
13.If a point is on the bisector of an angle, then it is equidistant from the sides of the
angle.
14.If a point in the interior of an angle is equidistant from the sides of that angle, then it
is on the bisector of the angle.
Postulate; Parallel Lines and Angles Formed
1. Through a point not on a given line, there is exactly one line parallel to the given
line.
2. If two lines are cut by a transversal so that alternate interior angles are congruent,
then the lines are parallel.
3. If two parallel lines are cut by a transversal, then the alternate interior angles are
congruent.
4. In a plane, if two lines are each perpendicular to a third line, then the two lines are
parallel.
5. If two parallel lines are cut by a transversal, then the interior angles on the same side
of the transversal are supplementary.
6. In a plane, if a line is perpendicular to one of two parallel lines, then it is
perpendicular to the other line also.
7. If two lines are cut by a transversal so that corresponding angles are congruent, then
the two lines are parallel.
8. If two parallel lines are cut by a transversal, then the corresponding angles are
congruent.
9. Two distinct lines parallel to the same line are parallel to each other.
10.If two lines are cut by a transversal so that interior angles on the same side of the
transversal are supplementary, then the lines are parallel.
11.The measure of an exterior angle of a triangle is greater than the measure of either of
its remote interior angles.
12.The sum of the measures of the angles of a triangle is 180°.
13.If two angles of one triangle are congruent to two angles of a second triangle, then
the third angles are congruent also.
14.A triangle can have no more than one right angle or one obtuse angle.
15.The acute angles of a right triangle are complementary.
16.The measure of each angle of an equilateral triangle is 60°.
17.If two triangles have a side and two angles of one congruent respectively to a side
and two angles of the other, then the triangle are congruent. AAS
18.The formula for the sum, S, of the measures of the interior angle of a polygon of n
sides is S = 180°(n – 2).
19.If a polygon has n sides and if all of its angles are congruent, then the formula for the
measure of one of its angles, a
–2)/n
20..21. The sum, E, of the measures of the exterior angles of a polygon taking one
exterior angle at each vertex, is 360°.
21.If a polygon of n sides is equiangular, then the measure e of one of its exterior angles
is e = 360°/ n .
Handout Page 7
Theorems; Quadrilaterals
1. A diagonal of a parallelogram forms two congruent triangles.
2. The opposite sides of a parallelogram are congruent.
3. The opposite angles of a parallelogram are congruent.
4. If two lines are parallel, then any two points on one of the lines are equidistant from
the other.
5. The diagonals of a parallelogram bisect each other.
6. Any two consecutive angles of a parallelogram are supplementary.
7. If two pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is
a parallelogram.
8. If two sides of a quadrilateral are parallel and congruent, then the quadrilateral is a
parallelogram.
9. The diagonals of a rectangle are congruent.
10. The diagonals of a rhombus are perpendicular to each other.
11. The diagonals of a rhombus bisect the angles of the rhombus.
12. If a segment joins the midpoint of two sides of a triangle, then it is parallel to the
third side and its length is one-half the length of the third side.
13. The median of a trapezoid is parallel to the bases and its length is one-half the sum
of the lengths of the bases.
14. If three or more parallel lines cut off congruent segments on one transversal, then
they cut off congruent segments on any transversal.
15. If a line is parallel to one side of a triangle and bisects the second side, then it bisects
the third side also.
16. If a line is parallel to the bases of a trapezoid and bisects one the non-parallel sides,
then it bisects the other side also.
17. If two sides of a triangle are not congruent, then the angles opposite these sides are
not congruent, and the smaller angle is opposite the smaller side.
18. If two sides of a triangle are not congruent, then the angles opposite these sides are
not congruent, and the smaller side is opposite the smaller angle.
19. The perpendicular segment from a point to a line is the shortest distance from the
point to the line.
20. The sum of the lengths of two sides of a triangle is greater than the length of the
third side.
21. If two sides of one triangle are congruent to two sides of another triangle and the
included angles are not congruent, then the remaining sides are also not congruent, and
the smaller side is opposite the smaller angle.
22. If two sides of one triangle are congruent to two sides of another triangle and the
third sides are not congruent, then the angles opposite the third sides are also not
congruent, and the smaller angle is opposite the smaller side.
23. All planes of the same dihedral angle are perpendicular.