1. GEOMETRY
Real
Numbers
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

2. GEOMETRY Natural Numbers Whole Numbers Integers Zero Negative Rational
Real Numbers
Natural
Numbers
Whole
Numbers
Integers
Zero
Negative
Integers
Rational
Numbers
Real
Numbers
Irregular
Fraction
Irrational
Numbers
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

3. Rational Numbers Irrational Numbers
GEOMETRY
Real Numbers
Rational Numbers
Rational Numbers = {p/q | p and q are integers and q ≠0}
Irrational Numbers
Special numbers like pi(∏) or √3
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

4. Integers Irregular Fraction GEOMETRY
Real Numbers
Integers
Integers = {…-3, -2, -1, 0, 1, 2, 3,…}
Irregular Fraction
Fractions that has “remainders” like 7/3, 9/5 or 3/2.
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

5. Whole Numbers Negative Integers GEOMETRY
Real Numbers
Whole Numbers
Whole Numbers = {0, 1, 2, 3,…}
Negative Integers
…-3, -2, -1
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

6. Natural Numbers Zero GEOMETRY Natural Numbers = {1, 2, 3,…}
Real Numbers
Natural Numbers
Natural Numbers = {1, 2, 3,…}
Zero
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

7. Real Number Properties
GEOMETRY
Real Numbers
Real Number Properties
Commutative
a + b = b + a a⋅ b = b ⋅ a
Associative
(a + b) + c = a + (b + c) (a ⋅ b) ⋅ c = a ⋅(b ⋅ c)
Distributive
a(b + c) = ab + ac
Identity
a + 0 = a a ⋅ 1 = a
Inverse
a + (-a) = 0 a ⋅ 1/a = 1
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

8. Equality Properties GEOMETRY Addition If a = b, then a + c = b + c
Real Numbers
Equality Properties
Addition
If a = b, then a + c = b + c
Multiplication
If a = b, then a ⋅ c = b ⋅ c
Reflexive
a = a
Symmetric
If a = b, then b = a
Transitive
If a = b and b = c, then a = c
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

9. Definition of Equivalence Relation
GEOMETRY
Real Numbers
Definition of Equivalence Relation
An equivalence relation is a relation that is
reflexive, symmetric, and transitive.
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

10. Order of Operations GEOMETRY Do the operations inside the parenthesis.
Real Numbers
Order of Operations
Do the operations inside the parenthesis.
PEMDAS:
Parenthesis
Exponent
Multiplication
Division
Addition
Subtraction
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

11. Segment Measure GEOMETRY
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

12. Ruler Postulate Every point of a line can be placed in
GEOMETRY
Segment Measure
Ruler Postulate
Every point of a line can be placed in
correspondence with a real number.
Note: the correspondence is in lowercase as it
represents the capital lettered point.
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

13. Definition of Distance
GEOMETRY
Segment Measure
Definition of Distance
The distance between two points
A and B is the absolute value of the
Difference of their coordinates.
Distance between points A and B is
Denoted by AB, given by AB = | a - b |
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

14. Definition of Segment’s Between
GEOMETRY
Segment Measure
Definition of Segment’s Between
A point M is between A and B if
AM + MB = AB.
The correct notation is A-M-B.
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

15. Completeness Postulate
GEOMETRY
Segment Measure
Completeness Postulate
Given a ray, AB, and any positive
real number r, there is exactly
one point C on the ray so that
AC = r. A B C r
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

16. Distance Formula For the distance between two points
GEOMETRY
Segment Measure
Distance Formula
For the distance between two points
on a Cartesian plane, the distance
formula is often used:
d = √ ((x1 - x2)2 + (y1 - y2)2)
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

17. GEOMETRY X1 = 3 Y1 = -1 X2 = -7 Y2 = 3 Segment Measure (-7,3) (0,0)
(3,-1)
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

18. Segment Bisectors GEOMETRY
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

19. Definition of a Midpoint
GEOMETRY
Segment Bisectors
Definition of a Midpoint
The midpoint of AB is M if A-M-B
And AM = MB.
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

20. Midpoint Theorem If M is the midpoint of AB, then AM = 1/2 AB.
GEOMETRY
Segment Bisectors
Midpoint Theorem
If M is the midpoint of AB, then
AM = 1/2 AB.
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

21. Definition of a Bisector
GEOMETRY
Segment Bisectors
Definition of a Bisector
A bisector of a segment is a curve
that intersects the segment only
at the midpoint.
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

22. Definition of Congruent Segments
GEOMETRY
Segment Bisectors
Definition of Congruent Segments
Congruent segments are segments
That have the same length.
The symbol ≅is used for
congruent segments
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

23. Perimeter and Circumference GEOMETRY
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

24. Definition of Perimeter
GEOMETRY
Perimeter and Circumference
Definition of Perimeter
Perimeter is the distance around
a closed curve.
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

25. Theorem 3.2 The perimeter of a regular n-gon with sides of length s is
GEOMETRY
Perimeter and Circumference
Theorem 3.2
The perimeter of a regular n-gon
with sides of length s is
n (number of sides) x s (sides).
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

26. Definition of Circumference
GEOMETRY
Perimeter and Circumference
Definition of Circumference
Circumference is the distance
around a circle.
Circumference = ∏ x d = 2 ∏ r
d = diameter r = radius
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

27. Polygons: Inscribed and Circumscribed GEOMETRY
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

28. Definition of an Inscribed Polygon
GEOMETRY
Polygons: Inscribed and Circumscribed
Definition of an Inscribed Polygon
An inscribed polygon is a polygon
whose vertices are points of a
circle.
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

29. Definition of Circumscribing Circle
GEOMETRY
Polygons: Inscribed and Circumscribed
Definition of Circumscribing Circle
A circumscribing circle is a circle
that surrounds and contains the
vertices of the polygon.
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

30. Definition of an Inscribed Angle
GEOMETRY
Polygons: Inscribed and Circumscribed
Definition of an Inscribed Angle
An inscribed angle is an angle whose
vertex is on a circle and whose sides
each contain another point on the
circle
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

31. Definition of Circumscribing Polygon
GEOMETRY
Polygons: Inscribed and Circumscribed
Definition of Circumscribing Polygon
A polygon circumscribed about a circle
is a polygon whose sides each intersect
the circle in exactly one point.
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

32. Definition of a Tangent Line
GEOMETRY
Polygons: Inscribed and Circumscribed
Definition of a Tangent Line
A tangent line (or tangent) is a line
in the plane of a circle that intersects
the circle in exactly one point.
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

33. Definition of a Tangency Point
GEOMETRY
Polygons: Inscribed and Circumscribed
Definition of a Tangency Point
The point of tangency is the point
at which a tangent line and a circle
intersect.
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

34. Definition of a Tangent Segment
GEOMETRY
Polygons: Inscribed and Circumscribed
Definition of a Tangent Segment
A tangent segment is a segment of a
tangent line that contains the point
of tangency.
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

35. Tangent Lines, Point of Tangency, and Tangent Segments GEOMETRY
Polygons: Inscribed and Circumscribed
Tangent Lines,
Point of Tangency,
and Tangent Segments
Joseph Angelo Tiongson. Copyright Discovery Christian Academy

36. Segment Constructions GEOMETRY
Joseph Angelo Tiongson. Copyright Discovery Christian Academy