2.5 Reasoning in Algebra and geometry

-Addition Property of Equality: If a = b, then a + c = b + c

-Subtraction Property of Equality: If a = b, then a – c = b – c

-Multiplication Property of Equality: If a = b, then a x c = b x c

-Division Property of Equality: If a = b and c ≠ 0, thena/c = b/c

2.5 Reasoning in Algebra and geometry

-Reflexive Property of Equality: a = a

-Symmetric Property of Equality: If a = b, then b = a

-Transitive Property of Equality: If a = b and b = c, then a = c

-Substitution Property of Equality: If a = b, then b can replace a in any expression

-Distributive Property of Equality: a(b + c) = ab + ac

2.5 Reasoning in Algebra and geometry

-Reflexive Property of Congruence:segment AB is congruent to segment ABangle A is congruent to angle A

Symmetric Property of Congruence:If segment AB is congruent to segment CD, then segment CD is congruent to segment ABIf angle A is congruent to angle B, then angle B is congruent to angle A

-Transitive Property of Congruence:If segment AB is congruent to segment CD and segment CD is congruent to segment EF, then segment AB is congruent to segment EFIf angle A is congruent to angle B and angle B is congruent to angle C, then angle A is congruent to angle C

2.6 Proving angles congruent

-Theorem 2.1: The vertical angle theorem

2.6 Proving angles congruent

-Paragraph Proof: a proof written as sentences in a paragraph.

Ex. Given angles 1 and 2 are vertical angles, we will prove that angle 1 and angle 2 are congruent. By the angle addition postulate, angle 1 plus angle 3 equals 180* and angle 2 plus angle 3 equals 180*. By substitution, angle 1 plus angle 3 equals angle 2 plus angle 3. Subtracting angle 3 from each side, you get angle 1 equals angle 2. We have proved the vertical angle theorem.

2.6 Proving angles congruent

-Theorem 2.2: congruent supplement theorem says if 2 angles are supplements of the same angle (or of congruent angles), then the 2 angles are congruent (proof similar to VAT)

-Theorem 2.3: congruent complement theorem says if 2 angles are complements of the same angle (or of congruent angles), then the 2 angles are congruent

-Theorem 2.4: all right angles are congruent

-Theorem 2.5: if 2 angles are congruent and supplementary, then each is a right angle

3.1 Lines and angles

-Parallel Lines: coplanar lines that do not intersect

-Parallel Planes: planes that do not intersect

-Skew Lines: noncoplanar lines that do not intersect

3.1 Lines and angles

-Transversal: a line that intersects two coplanar lines at 2 distinct points

-Alternate Interior: angles 4 and 6

-Same Side Interior: angles 4 and 5

-Corresponding: angles 4 and 8

3.2 Properties of Parallel Lines

If a transversal intersects two parallel lines, then…

-Postulate 3.1: Corresponding angles Postulate says corresponding angles are congruent

-Theorem 3.1: Alternate Interior anglesTheorem says Alt Int angles are congruent

-Theroem 3.2: Same Side Interior angles Theorem says same side Int angles are supplementary

-Theorem 3.3: Alternate Exterior angles Theorem says alt ext angles are congruent (angles 1 and 8)

-Theorem 3.4: Same Side Exterior angles Theorem says same side ext angles are supplemntary (angles 1 and 7)

3.3 Proving Lines Parallel-Postulate 3.2: Converse of the Corresponding Angles Postulate says if 2 lines and a transversal form corresponding angles that are congruent, then the 2 lines are Parallel

-Theorem 3.5: Converse of the Alternate Interior angles Theorem says if 2 lines and a transversal form Alt Int angles that are Congruent, then the 2 lines are parallel

-Theroem 3.2: Converse of the Same Side Interior angles Theoremsays if 2 lines and a transversal form Same Side Int angles that are Supplementary, then the 2 lines are parallel

3.3 Proving Lines Parallel

-Theorem 3.7: Converse of the Alternate Exterior angles Theorem says if 2 lines and a transversal form Alt Ext angles that are Congruent, then the 2 lines are parallel

-Theroem 3.8: Converse of the Same Side Exterior angles Theorem says if 2 lines and a transversal form Same Side Ext angles that are Supplementary, then the 2 lines are parallel

3.4 Parallel and Perpendicular Lines

-Theorem 3.9: If 2 lines are parallel to the same line, then they’re parallel to each other

-Theorem 3.10: In a plane, if 2 lines are perpendicular to the same line, then they’re parallel to each other

-Theorem 3.11: In a plane, if a line is perpendicular to 1 of 2 parallel lines, then it is also perpendicular to the other

3.5 Parallel lines and triangles

-Theorem 3.12: The angles of a triangle sum to 180 degrees

-Exterior Angle of a Polygon: An angle formed by a side and an extension of an adjacent side

-Remote Interior Angles: the twoNon-adjacent interior angles

-Theorem 3.13: The Triangle ExteriorAngle theorem says the measure of Each exterior angle equals the sum of the measures of the two remote interior angles

3.8 Slopes of Parallel and Perpendicular Lines

-Slopes of Parallel Lines: 1) If 2 nonvertical lines are parallel, then their slopes

are equal2) If the slopes of 2 distinct nonvertical lines are equal,

the lines are parallel3) Any 2 vertical lines are parallel

-Slopes of Perpendicular Lines: 4) If 2 nonvertical lines are perpendicular, the product

of their slopes is -1 5) If the slopes of 2 lines have a product of -1, the lines

are perpendicular6) Any horizontal line and vertical line are perpendicular

4.1 Congruent Figures

-Congruent Polygons: have corresponding angles and sides

-Theorem 4.1: If 2 angles of 1 triangle are congruent to 2 angles of another triangle, then the 3rd angles are congruent

4.2 Triangle Congruence by SSS and SAS

-Postulate 4.1: Side Side Side (SSS) congruence says if 3 sides of 1 triangleare congruent to 3 sides of another triangle, then the 2 triangles are congruent

-Postulate 4.2: Side Angle Side (SAS) congruence says if 2 sides and the included angle of 1 triangle are congruent to 2 sides and theincluded angle of another triangle, then the 2 triangles are congruent

4.3 Triangle Congruence by ASA and AAS

-Postulate 4.3: (ASA) congruence says if 2 angles and the included side of 1 triangleare congruent to 2 angles and theincluded side of another triangle,then the 2 triangles are congruent

-Postulate 4.4: Angle Angle Side (AAS) congruence says if 2 angles and the non-included side of 1 triangle are congruent to 2 angles and thenon-included side of another triangle, then the 2 triangles are congruent

4.4 corresponding Parts of Congruent Triangles

Triangles can be proved congruent by:

SSS - SAS - ASA - AAS