Proof definitions, postulates and theorems Flashcards
Addition Poe
If a = b, then
a+c=b+c
Perpendicular Lines
Meet at a right angle
Equilateral
A polygon where all sides are congruent
Equiangular
All angles are congruent
Subtraction Poe
If a = b, then
a-c=b-c
Multiplication Poe
If a = b, then
a(c)=b(c)
Division Poe
If a = b, then
a/c=b/c
Substitution Poe
If a = b, then
“a” can be substituted for “b” in any expression
Distributive Poe
If a = b, then
a(b+c)+ab+ac
Reflexive Poe
Basically, a thing equals itself
Symmetric Poe
Basically, if one expression equals another it doesn’t matter which expression goes on which side.
Transitive Poe
Basically, if two things equal a third thing, they also equal each other
Reflexive Poc
Basically, a thing(angle or side) is congruent to itself
Symmetric Poc
Basically, if one thing(angle or side) equals another it doesn’t matter which thing goes on which side
Transitive Poc
Basically, if two thing(angle or side) equal a third thing, they also equal each other
Right Angles Congruence Theorem
All right angles are congruent
Vertical Angles Congruence Theorem
All vertical angles are congruent
Linear Pair Postulate
Angles in a linear pair are supplements
Congruent Complements Theorem
If two angles are complementary to the same angle, they are congruent to each other
Congruent Complements Theorem
If two angles are supplementary to the same angle, they are congruent to each other
Parallel Postulate
given a line and a point not on the line, you can draw only one line that is parallel to the original line and goes through the point
Perpendicular Postulate
given a line and a point not on the line, you can draw only one line that is perpendicular to the original line and goes through the point
Transversal
a line that intersects two or more other lines at different points
Consecutive Interior Angles
Angles inside the lines and on the same side of the transversal
Alternate Interior Angles
Angles inside the lines and on opposite sides of the transversal
Alternate Exterior Angles
Angles outside the lines and opposite of the transversal
Corresponding Angles
Same corner of different intersections
Consecutive Interior Angles Postulate
For parallel lines crossed by a transversal, interior angles are supplementary
Alternate Interior Angles Postulate
For parallel lines crossed by a transversal, alternate interior angles are congruent
Alternate Exterior Angles Postulate
For parallel lines crossed by a transversal, alternate exterior angles are congruent
Corresponding Angles Postulate
For parallel lines crossed by a transversal, corresponding angles are congruent
If parallel lines are cut by a transversal…
- Consecutive interior angles are supplementary
- Alternate interior, alternate exterior and corresponding angles are congruent
Consecutive Interior Angles Converse Theorem
For lines cut by a transversal, if their consecutive interior angles are congruent then the lines are parallel
Alternate Interior Angles Converse Theorem
For lines cut by a transversal, if their alternate interior angles are congruent then the lines are parallel
Alternate Exterior Angles Converse Theorem
For lines cut by a transversal, if their alternate exterior angles are congruent then the lines are parallel
Corresponding Angles Converse Theorem
For lines cut by a transversal, if their corresponding angles are congruent then the lines are parallel
Transitive Property of Parallel Lines
If a is parallel to b and b is parallel to c, then a is parallel to c
Linear Pair Perpendicular Theorem
If two lines intersect to form a pair of congruent angles, then the lines are perpendicular
“Four Right Angles” Theorem
If two lines are perpendicular, then they intersect to form 4 right angles
“Perpendicular Sides Complementary Angles” Theorem
If two sides of two adjacent angles are perpendicular, then the angles are complementary
Perpendicular Transversal Theorem
If a transversal is perpendicular to one of the two parallel lines, then it is perpendicular to the other
Lines perpendicular to the Transversal Theorem
In a plane if two lines are perpendicular to the same lines, then they are parallel to each other
Triangle Sum Theorem
The angles of a triangle add up to 180 degrees
Exterior Angles Theorem
The measure of an exterior angle equals the sum of the remote interior angle(the two interior angles NOT adjacent to the exterior angle)
Definition of Congruent Triangles
If two triangles have 3 pairs of congruent corresponding angles and 3 pairs of congruent corresponding sides, the triangles are congruent
CPCTC
Corresponding parts of congruent triangles are congruent
SSS Postulate
If two triangles have 3 pairs of congruent corresponding sides, then the triangles are congruent
SAS Postulate
If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle then the two triangles are congruent
Hypotenuse Leg Congruence Postulate
If the hypotenuse and leg of a right triangle are congruent to the hypotenuse and leg of another right triangle, the triangles are congruent
ASA Postulate
If two angles and the included side(side in between two angles) of two triangles are congruent then the triangles are congruent
AAS Postulate
If two angles and a non included side of two triangles are congruent then the triangles are congruent
Base Angle Theorem
If two sides of triangles are congruent, then the angles opposite from them are congruent
Converse of Base Angle Theorem
If two angles of a triangle are congruent, then the sides opposite from them are congruent
Corollaries to Base Angle Theorem
If a triangle is equilateral, it is equiangular and if a triangle is equiangular, it is equilateral
Midsegment Theorem
The segment connecting the midpoints of two sides of a triangle(the midsegment) is parallel to and half as long as the third side
Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment
Converse of Perpendicular Bisector Theorem
If a point is equidistant from the endpoints of a segment,, it is on the perpendicular bisector of a segment
Angle Bisector Theorem
If a point is on the angle bisector of an angle then it is equidistant from the two sides of the angle
Converse of Angle Bisector Theorem
If a point inside an angle is equidistant from the two sides of the angle then it is on the angle bisector
Altitude
A line segment that connects a vertex of a triangle with the opposite side and is perp to that side
Median
A line segment that connects a vertex of a triangle with the midpoint of the opposite side
Triangle Inequality Theorem
Basically, if the third side is too short or too long, we end up with a line and not a triangle
“All Lengths are Similar” Theorem
If two shapes are similar, all of their length-related measures are the same ratio as the scale factor
AA Similarity Postulate
if two sets of corresponding angles of triangles are congruent, then the triangles are similar
SSS Similarity Theorem
If the ratios of three pairs of corresponding sides are equal, then the triangles are similar
SAS Similarity Postulate
If the ratio of two pairs of corresponding sides are equal AND the included angles are congruent, then the triangles are simialr
Side Splitter Theorem
If a line parallel to one side of a triangle intersects the two other sides, then it divides the sides proportionally
Extension of Side Splitter Theorem
If three parallel lines intersect two transversals then they divide the transversals proportionally
Angle Bisector Theorem(for triangles)
If a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to their adjacent sides