**Shapes**

When two pairs of parallel lines intersecting each others as left diagram.

If length of lines ab, bc, cd & ad are equal as well as the angles of intersection are right angles.

The four line form a **square**.

If line bc shift to the right and we have length of line ab equal line dc and length of line ad equal to line bc. If the angles of intersection still remain as right angles. The four lines form a **rectangle**.

If the pairs of parallel lines intersecting each others with angles not equal to right angle, they form a **parallelogram**.

Properties of parallelograms:

Line ab parallel to line dc (ab // dc) and both lines same in length (ab = dc)

Line ad parallel to line bc (ad // bc) and both lines same in length (ad = bc)

∠ adc = ∠ abc, ∠ dab = ∠ dcb

∠ adc + ∠ abc + ∠ dab + ∠ dcb = 360^{0}

If the pairs of parallel lines are equal in length but the intersection angles not equal to right angle, they form a **rhombus.**

ab // cd, ac // bd, ab = cd = ac = bd

and ∠ cab = ∠ cdb, ∠ acd = ∠ abd

The bisectors of the interior angles perpendicular to each others.

Since a square, rhombus and rectangle are formed by two set of parallel lines, they are also parallelograms.

A **quadrilateral** has 4 sides, 4 coners and the interior angles sum up to 360 degrees.

So a square, a rhombus, a rectangle and a parallelogram are also specific types of **quadrilateral**.

Only a square is regular quadrilateral and all other quadrilaterals are irregular.

A **trapezoid** is a quadrilateral that has one pair of opposite sides is parallel.

ab // dc, sum of all interior angles = 360^{0}

If a trapezoid is called an isosceles trapezoid, the sides that aren’t parallel are equal in length and both angles coming from a parallel side are equal, as shown.

ab // dc, ad = bc

and ∠ dab = ∠ abc, ∠ adc = ∠ bcd

sum of all interior angles = 360^{0}

Three straight lines intersecting each others can form a disclosed and it is called a **triangle**.

The sum of all interior angles of a triangle is 180 degrees

∠ a + ∠ b + ∠ c = 180^{0}

To proof:

Add a dot line as left figure shown and it is parallel

to one side (line x) of the triangle.

We have ∠ b = ∠ d, ∠ c = ∠ e and ∠ a + ∠ d + ∠ e = 180^{0 }(straight angle)

Therefore, ∠ a + ∠ b + ∠ c = 180^{0}, which is the sum of all three interior angles of the triangle.

You can also see that ∠ f = ∠ a + ∠ e = ∠ a + ∠ c, another word, the exterior angle ∠ f of ∠ b is the sum of the interior angles ∠ a and ∠ c of the triangle.

One of the three interior angles of a triangle is right angle (90^{0}), it is a right angle triangle.

∠ b = 90^{0}

∠ b + ∠ a + ∠ c = 180^{0}

∠ a + ∠ c = 90^{0}

A triangle with two of it sides having the same length, it is called isosceles triangle.

line ab = line ac

We can see that ∠ abc = ∠ acb

and the dotted line bisecting ∠ bac into half.

A triangle with two of the sides same in length and one of the interior is right angle, it is called isosceles right angle triangle.

Line ab = line bc

and ∠ abc = 90^{0} (right angle)

We can see that ∠ bac = ∠ acb = 45^{0}

A triangle that with all three sides same in length, it is call an equilateral triangle.

ab = ac = bc

It three interior angles measure equal in degrees.

Since sum of all three interior angles of triangle = 180^{0}

∠ abc = ∠ acb = ∠ bac = 60^{0}

Both triangle and quadrilateral are **polygon**. A triangle is a 3-sided polygon and a quadrilateral is a 4-sided polygon.

A **pentagon** is a 5-sided polygon with all sides equal in length.

A pentagon has 5 interior angles.

You can see that a pentagon can form by three triangles.

Since total interior angles measure 180^{0} for one triangle,

Three triangles give total angles of 3 × 180 = 540^{0}

The interior angles are equal, so every interior angle for pentagon measure 540 ÷ 5 = 108^{0}

A **hexagon** is a 6-sided polygon with all sides equal in length.

A hexagon has 6 interior angles.

You can see that a hexagon can form by four triangles.

Since total interior angles measure 180^{0} for one triangle,

Three triangles give total angles of 4 × 180 = 720^{0}

The interior angles are equal, so every interior angle for pentagon measure 720 ÷ 6 = 120^{0}

What is the interior angle measured for an **octagon**?

An octagon has 8 equal sides and 8 equal interior angles.

Total interior angle = 6 × 180 = 1080^{0}

Each interior angle = 1080 ÷ 8 = 135^{0}

Shapes of polygon | No. of sides | No. of interior angles | No. of exterior angles | Sum of interior angles | Special shapes | Each interior angle measured |

Triangle | 3 | 3 | 3 | 180^{0} |
Equilateral triangle | 60^{0} |

4-sided | 4 | 4 | 4 | 360^{0} |
Square | 90^{0} |

5-sided | 5 | 5 | 5 | 540^{0} |
Pentagon | 108^{0} |

6-sided | 6 | 6 | 6 | 720^{0} |
Hexagon | 120^{0} |

7-sided | 7 | 7 | 7 | 900^{0} |
Heptagon | 128.6^{0} |

8-sided | 8 | 8 | 8 | 1080^{0} |
Octagon | 135^{0} |

**Solids**

A square has height and width. It is called a two dimensional (2D) object, if we add a depth to the square, it gives us a **solid** object or 3 dimensional (3D) object. 3D objects is what we see in the real world.

In general, the square is a plane or face; it can be in various shapes. Examples: a rectangle, a pentagon, a circle etc…

A **cube** is a 3D object that have 6 faces (surfaces) and each face is a square same in size. Each face has 4 edges, 12 edges, 8 Vertices (corner points) and at each vertex 3 edges meet.

A cube is a special shape of **cuboid**.

A **cylinder** is a 3D solid object that has 3 faces, both end are circles in shape.

A **cone** is a 3D solid object that has 2 faces.

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