SOLID GEOMETRY

1. INSTRUCTOR: ENGR. MA. JESSA P. DOLLADO, MP

3. INTRODUCTION TO POINTS, LINES,
PLANES, AND ANGLES
• POINT – is a location in space; it indicates position. It occupies no space of it’s own,
and it has no dimension of it’s own.

4. INTRODUCTION TO POINTS, LINES,
PLANES, AND ANGLES
• MIDPOINT – is the point exactly halfway between two endpoints of a line segment.

5. INTRODUCTION TO POINTS, LINES,
PLANES, AND ANGLES
• COLLINEAR POINTS – are points that form a single straight line when they are
connected (two points are always collinear).
• NONCOLLINEAR POINTS – are points that do not form a single straight line when
they are connected (only three or more points can be noncollinear).

6. INTRODUCTION TO POINTS, LINES,
PLANES, AND ANGLES
• COPLANAR POINTS – are points that occupy the same plane.
• NONCOPLANAR POINTS – are points that do not occupy the same plane.

7. INTRODUCTION TO POINTS, LINES,
PLANES, AND ANGLES
• LINE – is a set of continuous points infinitely extending in opposite directions. It has
infinite length, but no depth or width.

8. INTRODUCTION TO POINTS, LINES,
PLANES, AND ANGLES
• RAY – begins at a point (called an endpoint because it marks the end of a ray), and
definitely extends in one direction.

9. INTRODUCTION TO POINTS, LINES,
PLANES, AND ANGLES
• LINE SEGMENT - is part of a line with two endpoints.

10. INTRODUCTION TO POINTS, LINES,
PLANES, AND ANGLES
• INTERSECTING LINES – are two or more different lines that meet at the same point.
• TRANSVERSAL LINES – is a line that cuts across two or more lines.

11. INTRODUCTION TO POINTS, LINES,
PLANES, AND ANGLES
• PARALLEL LINES – are coplanar lines that never intersect; they travel similar paths
at a constant distance from one another.
• PERPENDICULAR LINES – are lines, segments or rays that intersect to form right
angles. The symbol ⊥ means is perpendicular to .

12. INTRODUCTION TO POINTS, LINES,
PLANES, AND ANGLES
• SKEW LINES – are noncoplanar lines that never intersect; they travel dissimilar
paths on separate planes.

13. INTRODUCTION TO POINTS, LINES,
PLANES, AND ANGLES
• ANGLE – is a space formed by two rays called sides sharing a common endpoint
called vertex.

14. INTRODUCTION TO POINTS, LINES,
PLANES, AND ANGLES
• ANGLE MEASUREMENT:
1. DEGREE – (° or deg) is defined as the unit of angle measurement wherein one
complete revolution is divided into 360 parts.
2. RADIAN – (rad) is defined as the unit of angle measurement wherein one
complete revolution is equal to 2𝜋.
3. GRADIENT – (grad) is defined as the unit of angle measurement wherein one
complete revolution is divided into 400 parts.
4. MIL – (mil) is defined as the unit of angle measurement wherein one complete
revolution is divided into 6400 parts.

15. INTRODUCTION TO POINTS, LINES,
PLANES, AND ANGLES
• ADJACENT ANGLE – share a vertex, a side, and no interior points; they are angles
that lie side-by-side.
• ANGLE BISECTOR – is a ray that divides an angles into two equal angles.

16. INTRODUCTION TO POINTS, LINES,
PLANES, AND ANGLES
• Two angles are called COMPLEMENTARY ANGLES if the sum of their degree
measurements equals 90 degrees.
• Two angles are called SUPPLEMENTARY ANGLES if the sum of their degree
measurements equals 180 degrees.

17. INTRODUCTION TO POINTS, LINES,
PLANES, AND ANGLES
• When straight lines intersect, opposite angles, or angles nonadjacent to each other,
are called VERTICAL ANGLES. They are always congruent.

18. BASIC CONCEPTS & FORMULAS OF
POLYGONS
• POLYGON – is a closed figure made up of line segments (not curves) in a two-
dimensional plane. Polygon is the combination of two words, i.e. poly (means many)
and gon (means sides). A minimum of three line segments is required to connect
end to end, to make a closed figure.

19. PARTS OF POLYGON
1. SIDE or EDGE – is one of the line segments that make up the polygon.
2. VERTEX – is a point where the sides meet.
3. DIAGONAL – is a line connecting two non-adjacent vertices.
4. INTERIOR ANGLE - is the angle formed by two adjacent sides inside the polygon.
5. EXTERIOR ANGLE - an angle formed outside a polygon by one side & an extension of an
adjacent side; the supplement of an interior angle of the polygon.
6. CENTRAL ANGLE – (of a regular polygon) is the angle subtended by a side about the
center.
7. APOTHEM - (of a regular polygon) is the segment connecting the center of a polygon and
the midpoint of a side.

20. PARTS OF POLYGON

21. SIMILAR POLYGONS
Polygons are said to be SIMILAR if their corresponding interior angles are equal and
their corresponding sides are proportional.
𝐴1
𝐴2
= (
𝐴𝐵
𝑊𝑋
)2

22. BASIC CONCEPTS & FORMULAS OF
POLYGONS
SIMPLE POLYGON – a simple polygon that has only one boundary and the sides do
not cross each other, otherwise, it is a COMPLEX POLYGON.
CONVEX POLYGON – has no internal angle more than 180° and if there are any
internal angle greater than a straight angle, then it is a CONCAVE POLYGON.

23. BASIC CONCEPTS & FORMULAS OF
POLYGONS
If all the polygon sides and interior angles are equal, then they are known as
regular polygons. The examples of regular polygons are square, equilateral triangle,
etc. In regular polygons, not only are the sides congruent but so are the angles. That
means they are equiangular. If otherwise, the polygon is said to be irregular.
• Properties of Regular Polygons
1. All its sides are equal.
2. All its interior angles are equal.
3. The sum of its exterior angles is 360°.

24. BASIC CONCEPTS & FORMULAS OF
POLYGONS

25. FORMULAS IN POLYGONS
PERIMETER – is the length around the
boundary of a closed two-dimensional region.
AREA – is the amount of material that would
be needed to cover a surface completely.
• Perimeter of regular polygon,
𝑷 = 𝒔𝒏
Where: n = number of sides
s = measure of one side
• Area of regular polygon,
A=
𝟏
𝟐
Pa or A=
𝒔𝟐𝒏
𝟒𝐭𝐚𝐧(
𝟏𝟖𝟎°
𝒏
)
Where: P = perimeter
a = apothem
s = measure of one side
n = number of sides
• Apothem of a regular polygon,
a=
𝒔
𝟐𝐭𝐚𝐧(
𝟏𝟖𝟎°
𝒏
)
s = measure of one side
n = number of sides

26. FORMULAS IN POLYGONS
PERIMETER – is the length around the
boundary of a closed two-dimensional region.
AREA – is the amount of material that would
be needed to cover a surface completely.
• Perimeter of regular polygon,
𝑷 = 𝒔𝒏
Where: n = number of sides
s = measure of one side
• Area of regular polygon,
A=
𝟏
𝟐
Pa or A=
𝒔𝟐𝒏
𝟒𝐭𝐚𝐧(
𝟏𝟖𝟎°
𝒏
)
Where: P = perimeter
a = apothem
s = measure of one side
n = number of sides
• Apothem of a regular polygon,
a=
𝒔
𝟐𝐭𝐚𝐧(
𝟏𝟖𝟎°
𝒏
)
s = measure of one side
n = number of sides