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SUBJECT: M103- PLANE & SOLID GEOMETRY
ACTIVITY TITLE: POLYGONS
LEARNING TARGETS: At the end of the lesson the learner will be able to:
- Distinguish between convex and concave polygons.
- Classify polygons according to side.
- Compute the interior angle sum of convex polygon and the degree measure of each
regular polygon.
- Recognize special quadrilaterals and special parallelograms.
- Apply various tests to determine whether a specific quadrilateral is a parallelogram.
- Enumerate the properties of each quadrilaterals and parallelograms.
- Apply the Midpoint Theorem for Triangles.
CLASSIFYING POLYGONS
Polygons
Polygon is basically any closed shape which is formed with three or more straight lines. There is no limit in how
many sides will be there in a polygon, it can possess infinite sides, they can have 10 sides or more sides.
Trigons
Trigons are polygons who have three sides. They are nothing but triangles. These trigons or triangles are further
classified into different categories, such as:
Scalene Triangle: All sides are unequal
Isosceles Triangle: Two sides are equal
Regular Polygon
In a regular polygon, all the sides of the
polygon are equal, and all the interior angles
are the same.
Examples:
A square has all its sides equal to
5cm, and all the angles are at 90°.
An equilateral triangle has all three
sides equal to 10cm and angles
measure to 60°.
A regular pentagon has 5 equal
sides and all the interior angles
measures to 108 degrees.
Irregular Polygon
A polygon with an irregular shape. It
means the sides and angles of the polygon
are not equal.
Example:
A quadrilateral with unequal sides.
An isosceles triangle has only two
of its sides equal, and the third side
has a different measurement.
Convex Polygon
In a convex polygon, the measure of the interior
angle is less than 180 degrees. It is exactly
opposite to the concave polygon. The vertices of
a convex polygon are always outwards.
Example : See the figure of an irregular hexagon,
whose vertices are outwards.
Concave Polygon
In a concave polygon, at least one angle
measures more than 180 degrees. The
vertices of a concave polygon are inwards
as well as outwards.
Equilateral Triangle: All the three sides are equal and all angles measures to 60 degrees.
ANGLE SUM OF POLYGONS
Sum of angles of a polygon = (n-2) x 180 °
Let’s look at some example
SHAPES NO. OF SIDES SUM OF THE ANGLES
3 (3-2) = 1x
4 (4-2)= 2x 180 ° = 360 °
5 (5-2)= 3x 180 ° = 540 °
Quadrilateral Polygon
Quadrilateral polygon is also called a four-
sided polygon or a quadrangle. The different
types of the quadrilateral polygon are square,
rectangle, rhombus and parallelogram.
Pentagon Polygon
The five-sided polygon is called pentagon polygon.
When all the five sides of the polygon are equal in
length, then it is called regular pentagon otherwise
irregular pentagon.
Hexagons
Another type of polygon is the hexagon which
has 6 sides and 6 vertices. A regular hexagon
will have equal 6 sides and all its interior and
exterior angles also measure equals.
Equilateral Polygons
The polygons whose all the sides are equal
are called equilateral polygons, for example,
an equilateral triangle, a square, etc.
Equiangular Polygons
The polygons whose all the interior angles
are equal such as a rectangle are called
equiangular polygons.
c. Proving that figures are Parallelogram.
How do we prove a quadrilateral is a parallelogram?
Six basic properties of parallelograms to be true!
- Both pairs of opposite sides are parallel
- Both pairs of opposite sides are congruent
- Both pairs of opposite angles are congruent
- Diagonals bisect each other
- One angle is supplementary to both consecutive angles (same-side interior)
- One pair of opposite sides are congruent and parallel
PROPERTIES OF SPECIAL PARALLELOGRAMS
A parallelogram is a plane figure with two pairs of opposite sides. The opposite sides are parallel and equal, and
the opposite angles are of equal measure. Parallelograms can be equilateral, equiangular, or both. There are three
special types of parallelograms— rectangle, rhombus, and square. They are special because, in addition to the
general properties of a parallelogram that they show, they have their unique properties. The unique properties are
as follows:
A rectangle has four right angles. So it is equiangular (with all angles equal)
A rhombus has four congruent sides. So it is equilateral (all sides equal)
A square has four right angles and four congruent sides. So a square is equilateral and equiangular.
Properties
Let us consider each of the properties of special parallelograms in the following segments.
RECTANGLE
- Opposites sides are equal
- Opposite sides are parallel
- All angles are equal & 90 °
- Diagonals equal & bisects each other
SQUARE
- Opposites sides are equal
- Opposite sides are parallel
- All angles are equal & 90 °
- Diagonals equal & perpendicular bisectors
RHOMBUS
- Opposites sides are equal
- Opposite sides are parallel
- All angles are equal & 90
- Diagonals equal & perpendicular bisectors
MID-POINT THEOREM
The midpoint theorem states that “ The line segment in a triangle joining the midpoint of any two
sides of the triangle is said to be parallel to its third side and is also half of the length of the third side .”
Mid-Point Theorem Proof
If a line segment adjoins the mid-point of any two sides of a triangle, then the line segment is said to be parallel to
the remaining third side and its measure will be half of the third side.
Consider the triangle ABC, as shown in the above figure,
Let E and D be the midpoints of the sides AC and AB. Then the line DE is said to be parallel to the side BC,
whereas the side DE is half of the side BC; i.e., DE || BC
DE = (1/2 * BC).
Given: ∆ CFE ∧ ∆ ADE
Extend the line segment DE and produce it to F such that, EF = DE and join CF.
EC = AE
STATEMENT REASONS1.
- EC = AE 1. Given
2. ∠CEF = ∠AED
- vertically opposite angles
- DE = EF 3. By construction
4. ∠CFE = ∠ADE
- by c.p.c.t.
- ∠FCE= ∠DAE and CF = AD
- by c.p.c.t.
- ∠CFE and ∠ADE
- are the alternate interior angles
7. △ CFE ≅ △ ADE
- Given
- By SAS congruence
Assume CF and AB as two lines which are intersected by the transversal DF.
In a similar way, ∠FCE and ∠DAE are the alternate interior angles.
Assume CF and AB are the two lines which are intersected by the transversal AC.
Therefore, CF ∥ AB
So, CF ∥ BD
and CF = BD {since BD = AD, it is proved that CF = AD}
Thus, BDFC forms a parallelogram.
By the properties of a parallelogram, we can write
BC ∥ DF
and BC = DF
BC ∥ DE
and DE = (1/2 * BC).
Hence, the midpoint theorem is proved.
Example:
In triangle ABC, the midpoints of BC, CA, and AB are F, E, and D, respectively. Find the value of DE,
if the value of BC = 14 cm
