Download Postulates: 1) Reflexive Property of Equality/ Congruence

Postulates:
1) Reflexive Property of Equality/ Congruence
- a=a
2) Symmetric Property of Equality/ Congruence
- If a = b, then b = a.
3) Transitive Property of Equality/ Congruence
- If a = b, and b = c, then a = c.
4) Substitution Property of Equality/ Congruence
- If ab = 2ac, and ad = ac, then ab = 2ad.
5) Addition Property (Theorem) of Equality (Congruence)
- If a = b, and c = d, then a+c = b+d.
6) Subtraction Property (Theorem) of Equality (Congruence)
- If a = b, and c = d, then a-c = b-d.
7) Multiplication Property (Theorem) of Equality (Congruence)
- If a = b, and c = d, then ac = bd.
8) Division Property (Theorem) of Equality (Congruence)
- If a = b, and c = d, then a/c = b/d, where c ≠ 0 and d ≠ 0.
Inequality Postulates:
1) Addition/ Subtraction Property of Inequality
- If a ​
>​
b, then a+x ​
>​
b+x.
If a ​
>​
b, then a-x ​
>​
b-x.
- If a ​
<​
b, then a+x ​
<​
b+x.
If a ​
<​
b, then a-x ​
<​
b-x.
2) Positive Multiplication/ Division Property of Inequality
- If x ​
<​
y, and a​
>​
0, then ax ​
<​
ay.
If x ​
>​
y, and a​
>​
0, then ax​
>​
ay.
- If x ​
>​
y, and a ​
>​
0, then x/a ​
>​
y/a.
If x ​
<​
y, and a ​
>​
0, then x/a ​
<​
y/a.
3) Negative Multiplication/ Division Property of Inequality
- If x ​
<​
y, and a ​
<​
0, then ax​
>​
ay.
If x ​
>​
y, and a ​
<​
0, then ax ​
<​
ay.
- If x ​
>​
y, and a ​
<​
0, then x/a ​
<​
y/a.
If x ​
<​
y, and a ​
<​
0, then x/a​
>​
y/a.
4) Substitution Property of Inequality
- If a​
> b, and b ​
​
=​
c, then a > c.
- If a < b, and b ​
=​
c, then a < c.
5) Transitive Property of Inequality
- If a > b, and b ​
>​
c, then a > c.
- If a < b, and b ​
<​
c, then a < c.
6) A whole is greater than any of its parts.
7) Law of Trichotomy: Either a < b, a = b, or a > b.
8) The shortest path (distance) between a point and a line is the perpendicular segment from the
point to the line.
* Most inequality proofs are numerical.
Euclid’s 5th Postulate (The Parallel Postulate):
Given a line and a point not on the line, there exists one and only one (unique) line through the given
point that is parallel to the given line.
Theorems:
1) If 2 angles are right angles, then they are congruent.
2) If 2 angles are complementary (supplementary) to congruent angles/ the same angle, then these
angles are congruent.
3) If 2 adjacent angles form a right angle, then they are complementary.
4) If 2 angles form a linear pair, then they are supplementary.
* Never say: If a line segment is a segment bisector, then it divides the segment into 2 congruent
parts.
- Prove that point __ is the midpoint of the segment.
- Use the midpoint theorem to prove that it divides the segment into 2 congruent parts.
5) If 2 adjacent angles have noncommon sides that are opposite rays, then they are a linear pair of
angles.
6) If 2 angles are congruent to congruent/ the same angle, then they are congruent.
* Always use reflexive property to state what you are adding/subtracting: You need​
2 pairs ​
of
congruent things.
* Midpoints/ Bisectors: Think about utilising multiplication/ division theorem (property) of
congruence (equality)
* Multiplication (Division) Theorem of Congruence
If segments or angles are congruent, then their multiples (divisions) are congruent.
7) If 2 angles’ sides form 2 pairs of opposite rays, then they are vertical angles.
8) If 2 sides of a ∆are congruent, then the angles opposite these sides are also congruent.
9) If 2 angles of a ∆are congruent, then the sides opposite these angles are also congruent. (Converse
of 8)
10) If a segment is a median of a ∆, then it is drawn from any vertex of the ∆to the midpoint of the
opposite side.
11) If a segment is drawn from any vertex of a ∆to the midpoint of the opposite side, then it is a
median of the ∆. (Converse of 10)
12) If a segment is the altitude of a ∆, then it is drawn from any vertex of the ∆perpendicular to the
line containing the opposite side.
13) If a segment is drawn from any vertex of a ∆perpendicular to the line containing the opposite
side, then it is an altitude of the ∆. (Converse of 12)
* Must say: the ​
line containing the opposite side​
. An altitude could fall ​
outside​
the ∆.
* Essentially, a perpendicular bisector is both the median and the altitude.
Isosceles ∆s:
14) If a ∆has at least 2 congruent sides, then it is isosceles.
15) The median to the base, the altitude to the base, and the bisector of the vertex angle in an
isosceles ∆are all the same segment.
16) If a ∆is equilateral, then it is equiangular.
17) If a ∆is equiangular, then it is equilateral. (Converse of 16)
* All equilateral ∆s are isosceles, but not all isosceles ∆s are equilateral.
3 Equidistance Theorems:
18) If 2 points are each equidistant from the endpoints of a segment, then they will determine the
perpendicular bisector of that segment.
* ^: used only to establish perpendicular bisector
* Equidistant = congruent. No need to prove that segments are equal in measure in order to say that
they are equidistant.
19) If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints
of the segment.
20) If a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector
of the segment. (Converse of 19)
Circles:
21) If a segment is a radius, then it is drawn from the centre of a circle to any point on the circle.
22) If a segment is drawn from the centre of a circle to any point on the circle, then it is a radius.
(Converse of 21)
23) If 2 segments are of the same circle/ congruent circles, then they are congruent.
Proof by Contradiction/ Indirect Proof:
24) Law of Excluded Middle: Either p or ~p is true. No other possibilities exist.
25) Law of Contradiction: Both p and ~p cannot be true at the same time.
* Format:
- Step 1: Negation of what you are trying to prove: Law of Excluded Middle
- Step 2: Assumption of negation of given: Assumption
- Last step: Law of contradiction in steps __ and __. Assumption in step 2 is false. Negation of
assumption must be true.
Inequality Proofs:
26) The measure of the exterior angles of a ∆is greater than the measure of either remote interior
angle.
27) If 2 angles of a ∆are unequal, then the sides opposite them are unequal and the longer side is
opposite the larger angle.
28) If 2 sides of a ∆are unequal, then the angles opposite them are unequal and the larger angle is
opposite the longer side. (Converse of 28)
29) The sum of any 2 side lengths of a ∆is greater than the length of the third side.
Parallel Lines:
30) If 2 parallel lines are cut by a transversal, then each pair of:
- alternate interior angles/ alternate exterior angles/ corresponding angles is​
congruent
- same side interior angles/ same side exterior angles is ​
supplementary
31) If 2 coplanar lines are cut by a transversal such that a pair of:
- alternate interior angles/ alternate exterior angles/ corresponding angles is​
congruent
- same side interior angles/ same side exterior angles is ​
supplementary​
,
then the lines are parallel. (Converse of 30)
32) If 2 lines are parallel to a third line, then they are parallel to each other.
33) If 2 coplanar lines are perpendicular to the same line, then they are parallel.
∆Angle-Sum Theorems:
34) The sum of the measures of the interior angles of a ∆is 180°.
35) The acute angles of a right ∆are complementary.
36) No Choice Theorem: If 2 angles of one ∆are congruent to 2 angles of another ∆, then the third pair
of angles is congruent.
37) The measure of an exterior angle of a ∆is equal to the sum of the measures of its remote interior
angles.
∆Congruence:
38) If 2 ∆s are congruent, then all pairs of corresponding parts are congruent.
* No SSA ∆congruence!
39) SAS ​
Postulate​
: 2 sides + ​
included ​
angle of one ∆is congruent to corresponding sides and angle of
another ∆.
40) SSS ​
Theorem​
: All 3 sides of one ∆is congruent to corresponding sides of another ∆.
41) ASA ​
Theorem​
: 2 angles + ​
included ​
side of one ∆is congruent to corresponding angles and side of
another ∆.
42) AAA ​
Theorem​
: All 3 angles of one ∆is congruent to corresponding angles of another ∆.
43) HL Theorem - needs 3 things:
1. right ∆(not just right angle, you have to state that it is a right ∆),
2. Hypotenuses (side opposite right angle) are congruent,
3. One pair of legs (sides adjacent to right angle) are congruent.
* no need to state that right angles are congruent
44) If a ∆has a right angle, then it is a right ∆.
* The Transitive/ Substitution Property can apply to figures (polygons).
Polygons:
45) The sum of the measures of the interior angles of a n-gon is: (n-2) x 180°.
46) The sum of the measures of the exterior angles of a n-gon is: 360°.
47) The measure of each interior angle of an equiangular n-gon is: (n-2) x 180°/n.
48) The measure of each exterior angle of an equiangular n-gon is: 360°/n.
Parallelograms:
49) ​
Definition​
: A quadrilateral is a parallelogram ​
if and only if​
both pairs of opposite sides are parallel.
​
50) ​
Properties​
: If a quadrilateral is a parallelogram, then:
- both pairs of opposite sides/ opposite angles are congruent,
- any pair of consecutive angles are supplementary,
- the diagonals bisect each other​
.
51) ​
Proving a polygon is a parallelogram​
: If:
- both pairs of opposite sides of a quadrilateral are parallel (converse definition),
- one pair of opposite sides of a quadrilateral is both parallel and congruent,
- both pairs of opposite sides/ opposite angles of a quadrilateral are congruent,
- the diagonals of a quadrilateral bisect each other,
- an angle of a quadrilateral is supplementary to both of its consecutive angles,
then it is a parallelogram.
Rectangle:
52) ​
Definition​
: A quadrilateral is a rectangle ​
if and only if​
it is a ​
parallelogram​
with at least one right
angle.
53) ​
Properties​
: If a quadrilateral is a rectangle, then:
- it has all the properties of a parallelogram,
- it is ​
equiangular​
,
- the diagonals are congruent​
.
54) ​
Proving a polygon is a rectangle​
: If:
- a quadrilateral is a parallelogram with at least one right angle (converse definition),
- a quadrilateral is equiangular,
- a quadrilateral is a parallelogram with congruent diagonals,
then it is a rectangle.
Rhombus:
55) ​
Definition​
: A quadrilateral is a rectangle ​
if and only if​
it is a ​
parallelogram ​
with at least 2 congruent
consecutive sides.
56) ​
Properties​
: If a quadrilateral is a rhombus, then:
- it has all the properties of a parallelogram,
- it is ​
equilateral​
,
- the diagonals are perpendicular​
,
- each diagonal bisects a pair of opposite angles.
57) ​
Proving a polygon is a rectangle​
: If:
- a quadrilateral is a parallelogram with at least 2 congruent consecutive sides (converse
definition),
- a quadrilateral is a parallelogram whose diagonals are perpendicular,
- a quadrilateral is a parallelogram whose diagonals bisect a pair of opposite angles,
- a quadrilateral whose diagonals are perpendicular bisectors of each other,
- a quadrilateral is equilateral,
then it is a rhombus.
Square:
58) ​
Definition​
: A quadrilateral is a square ​
if and only if​
it is both a rectangle and a rhombus.
59) ​
Properties: ​
If a quadrilateral is a square, then:
- it has all the properties of a rectangle,
- it has all the properties of a rhombus.
(and thus has all the properties of a parallelogram)
60) ​
Proving a polygon is a square​
: If a quadrilateral is both a rectangle and a rhombus, then it is a
square (converse definition).
Trapezoid:
61) ​
Definition​
: A quadrilateral is a trapezoid ​
if and only if​
it has exactly one pair of parallel sides.
62) ​
Prove a polygon is a trapezoid​
: If a quadrilateral has exactly one pair of opposite sides that is
parallel, then it is a trapezoid (converse definition).
* parallel sides = bases; non-parallel sides = legs.
Isosceles Trapezoid:
63) ​
Definition​
: A quadrilateral is an isosceles trapezoid ​
if and only if​
it is a ​
trapezoid​
in which the
non-parallel sides/ legs are congruent.
64) ​
Properties​
: If a quadrilateral is an isosceles trapezoid, then:
- the lower base angles are congruent,
- the upper base angles are congruent,
- the diagonals are congruent​
.
65) ​
Prove a polygon is an isosceles trapezoid​
: If:
- a quadrilateral is a trapezoid with non-parallel sides/ legs that are congruent (converse
definition),
- a quadrilateral is a trapezoid with one pair of upper/lower base angles that is congruent,
- a quadrilateral is a trapezoid with congruent diagonals,
then it is an isosceles trapezoid.
Kite:
66) ​
Definition​
: A quadrilateral is a kite ​
if and only if​
it has 2 disjoint pairs of consecutive sides that are
congruent.
67) ​
Properties​
: If a quadrilateral is a kite, then:
- one of the diagonals is the perpendicular bisector of the other​
,
- the diagonals are perpendicular​
,
- one of the diagonals bisects a pair of opposite angles​
.
68) ​
Prove a polygon is a kite​
: If:
- a quadrilateral has 2 disjoint pairs of consecutive sides that are congruent (converse
definition),
- a quadrilateral has 1 diagonal that is the perpendicular bisector of the other diagonal,
then it is a kite.
Ratios and Proportions:
Ratio = quotient of 2 numbers
Proportion = an equation stating that 2 or more ratios are equal
Mean proportion = proportion in which the means are equal
69) In a proportion, the product of the means is equal to the product of the extremes. (Cross
Multiplication)
70) In a proportion, the means can be interchanged.
71) In a proportion, the extremes can be interchanged.
72) Midsegment (Midline) Theorem (of a ∆): A segment joining the midpoints of 2 sides of a ∆(called
the midsegment) is parallel to the third side and is ½ the length of the 3rd side.
73) Midsegment (Midline) Theorem (of a trapezoid): The median (midsegment) of a trapezoid is
parallel to the bases, and its length is ½ the sum of the lengths of the 2 bases.
* Format of 72: A segment joining the midpoints of 2 sides of a ∆is
- parallel to the 3rd side, OR
- ½ the length of the 3rd side.
* Format of 73:
- The median of a trapezoid is parallel to the bases, OR
- The length of the median of a trapezoid is ½ the sum of the lengths of the 2 bases.
74) Side-Splitter Theorem: If a line is parallel to 1 side of a ∆and intersects the other 2 sides, then it
divides those 2 sides proportionally.
75) If 3 or more parallel lines are intersected by 2 transversals, then the parallel lines divide the
transversals proportionally. (no need for similar ∆s)
76) Angle-Bisector Theorem: If a ray bisects an angle of a ∆, then it divides the opposite side into
segments that are proportional to the adjacent sides.
Right ∆s:
77) Right ∆s, Hypotenuse, Altitude: If an altitude is drawn to the hypotenuse of a right ∆, then:
a) the 2 ∆s formed are similar to the given right ∆and to each other,
b) either leg of the given ∆is the mean proportional between the hypotenuse of the given ∆and
the segment of the hypotenuse adjacent to that leg,
c) the altitude to the hypotenuse is the mean proportional between the segments of the
hypotenuse.
*
b) hypotenuse/leg 1 = leg 1/ adj. seg. 1
OR
hypotenuse/leg 2 = leg 2/ adj. seg. 2
c) adj. seg. 1/altitude = altitude/adj. seg. 2
78) Pythagorean Theorem: The square of the measure of the hypotenuse of a right ∆is = to the sum
of the squares of the measures of the legs.
Similar Polygons:
79) Ratio of Similitude/ Constant of Proportionality:
The ratio of the perimeters of 2 similar polygons is equal to the ratio of any pair of corresponding
sides.
80) If 2 polygons are similar, then:
- all pairs of corresponding angles are congruent,
- the corresponding sides are proportional.
Proving ∆s Similar:
81) AA~ Theorem: 2 angles of one ∆are congruent to corresponding angles of another ∆.
* All congruent ∆s are similar, but not all similar ∆s are congruent
82) SAS~ Theorem: 2 sides of one ∆is proportional to the corresponding sides of another ∆+ ​
included
angle of 1st ∆is congruent to 2nd ∆.
83) SSS ~ Theorem: All 3 sides of one ∆is proportional to corresponding sides of another ∆.
* Drawings are always dotted
Miscellaneous:
84) If 2 segments intersect each other at their midpoints, then they bisect each other.
85) If 2 segments bisect each other, then they intersect each other at their midpoints. (Converse of
83)