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
axc=bxc
-Division Property of Equality: If a = b and c ≠ 0, then
a/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 AB
angle 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 AB
If 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
EF
If 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 angles
Theorem 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 Theorem
says 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 two
Non-adjacent interior angles
-Theorem 3.13: The Triangle Exterior
Angle 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 equal
2) If the slopes of 2 distinct nonvertical lines are equal,
the lines are parallel
3) Any 2 vertical lines are parallel
-Slopes of Perpendicular Lines:
1) If 2 nonvertical lines are perpendicular, the product
of their slopes is -1
2) If the slopes of 2 lines have a product of -1, the lines
are perpendicular
3) 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 triangle
are 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 the
included 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 triangle
are congruent to 2 angles and the
included 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 the
non-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