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@Deyse SJ de Assis - Lisboa, Portugal

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Mathematics EN 9th grade: Polygons and Quadrilaterals

1. Polygons

A polygon is a flat figure formed by line segments.

✔ It has sides, vertices, and angles

✔ It is closed (forms a complete shape)

🔹 Interior angles

These are the angles inside the polygon.

📌 Sum of interior angles:

$ S = (n - 2)\cdot 180^\circ $

Where:

$ n $ = number of sides

📌 Example:

Pentagon ($ n = 5 $):

$ S = (5 - 2)\cdot 180 = 540^\circ $

🔹 Exterior angles

These are the angles formed outside the polygon.

📌 Important property:

The sum of exterior angles is always:

$ 360^\circ $

✔ Valid for any convex polygon

2. Quadrilaterals

They are polygons with 4 sides.

📌 Sum of interior angles:

$ 360^\circ $

3. Diagonals of a quadrilateral

A diagonal connects non-consecutive vertices.

✔ A quadrilateral always has 2 diagonals

4. Parallelogram

A quadrilateral with opposite sides parallel.

✔ Opposite sides are equal

✔ Opposite angles are equal

✔ Diagonals bisect each other

5. Rectangle

A parallelogram with right angles.

✔ All angles are $ 90^\circ $

✔ Diagonals are equal

6. Rhombus

A parallelogram with all sides equal.

✔ Diagonals are perpendicular

✔ Diagonals bisect the angles

7. Trapezoid

A quadrilateral with only one pair of parallel sides.

✔ Parallel bases

✔ Non-parallel sides are called legs

8. Kite (deltoid)

A quadrilateral with two pairs of consecutive equal sides.

✔ Diagonals are perpendicular

✔ One diagonal is an axis of symmetry

9. Important interpretation

✔ Polygons have interior angle sum given by $ (n - 2)\cdot 180^\circ $

✔ Quadrilaterals have sum $ 360^\circ $

✔ Diagonals help identify properties

✔ Each type of quadrilateral has specific characteristics

10. Key points for the exam

✔ Calculate sum of interior angles

✔ Know that exterior angles sum is $ 360^\circ $

✔ Identify types of quadrilaterals

✔ Recognize properties of diagonals

✔ Distinguish parallelogram, rectangle, rhombus, trapezoid, and kite

Did you know?

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Explore more about What are rational numbers?
They are numbers that can be written in fraction form:
$ \frac{a}{b}, \quad b \neq 0 $
Examples:
$ \frac{3}{4}, -2, 0{,}5, -\frac{7}{3} $
valor absoluto
The valor absoluto represents the distance to zero:
$ |a| = a $ se $ a \geq 0 $
$ |a| = -a $ se $ a
Examples:
\( |5| = 5 $
$ |-5| = 5 $
Simétrico
The Simétrico of a number is:
$ -a $
Examples:
$ 3 \rightarrow -3 $
$ -7 \rightarrow 7 $
Adição e subtração
Addition with the same sign
Add the values and keep the sign
Examples:
$ 3 + 5 = 8 $
$ -4 + (-6) = -10 $
Soma com sinais contrários
Subtract the values and take the sign of the number with the greatest absolute valueNL#
Examples:
$ 7 + (-3) = 4 $
$ -8 + 5 = -3 $
Soma de números simétricos
The result is always zero
$ a + (-a) = 0 $
Examples:
$ 5 + (-5) = 0 $
Subtração (diferença)
It is transformed into the sum of the opposite
$ a - b = a + (-b) $
Examples:
$ 7 - 3 = 7 + (-3) = 4 $
$ 5 - (-2) = 5 + 2 = 7 $
Divisão de números racionais
Main rule
Multiply by the reciprocal
$ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} $
Inverso de um número racional
Swap the numerator and denominator
$ \text{Inverso de } \frac{a}{b} = \frac{b}{a}, \quad a \neq 0 $
Examples:
$ \frac{3}{4} \rightarrow \frac{4}{3} $
$ 2 = \frac{2}{1} \rightarrow \frac{1}{2} $
Important:
$ 0 \text{ não tem inverso} $
Examples de divisão
$ \frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6} $
TIPS
✔ Same signs → add and keep the sign
✔ Different signs → subtract and take the sign of the number with the greatest absolute value
✔ Subtraction → convert to addition
✔ Division → multiply by the reciprocal
✔ Número + simétrico: $ a + (-a) = 0 $
✔ Valor absoluto: $ |a| \geq 0 $
..

What are rational numbers?
They are numbers that can be written in fraction form:
$ \frac{a}{b}, \quad b \neq 0 $
Examples:
$ \frac{3}{4}, -2, 0{,}5, -\frac{7}{3} $
valor absoluto
The valor absoluto represents the distance to zero:
$ |a| = a $ se $ a \geq 0 $
$ |a| = -a $ se $ a < 0 $
Examples:
$ |5| = 5 $
$ |-5| = 5 $
Simétrico
The ..know more

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Explore more about Word that replaces or accompanies a noun, avoiding repetition and indicating person, possession, quantity, or relation.
Ex.: ele, aquele, meu, ninguém.
Subclasses de pronomes
Pronome pessoal
Refers to the persons in speech: 1ª, 2ª ou 3ª pessoa.
- Formas tónicas: used alone or for emphasis (mim, ti, ele, nós).
- Formas átonas: used attached to the verb (me, te, se, o, a, nos, vos).
Pronome pessoal em adjacência verbal
- Ênclise: after the verb (amar-me-ás).
- Próclise: before the verb (me amarás).
- Mesóclise: in the middle of the verb, used with future tense (amar-me-ás).
Contração de pronomes pessoais
Personal pronouns (pronomes pessoais) can combine with direct (direto) and indirect (indireto) objects into a single form.
me + o = mo / nos + o = no=lo / vos + o = vo-lo
te + o = to / lhe + o = lho / lhes + o = lho
Ex.: Dei um telemóvel à minha irmã.
complemento direto + complemento indireto
Dei-o à minha irmã
Dei-lhe um telemóvel
Dei-lho
Pronome pessoal “se”
It can perform different syntactic functions or refer to certain phrasal structures.
- se reflexo: complemento direto (Ela penteia-se)
- se recíproco: complemento indireto (Eles abraçam-se)
- se impessoal: sujeito indeterminado (Diz-se que está a ficar cego)
- se passivo: partícula apassivante (Vendem-se maçãs.)
Pronome demonstrativo
Indicates position or reference: este, essa, aquele, isto, isso, aquilo.
Pronome possessivo
Indicates possession: meu, teu, seu, nossa, vossos.
Pronome indefinido
Indicates vague quantity or identity: alguém, ninguém, tudo, algum, certo.
Pronome relativo
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Word that replaces or accompanies a noun, avoiding repetition and indicating person, possession, quantity, or relation.
Ex.: ele, aquele, meu, ninguém.
Subclasses de pronomes
Pronome pessoal
Refers to the persons in speech: 1ª, 2ª ou 3ª pessoa.
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✔ Frase passiva: the subject receives the action.
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👉 Tip:
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CD da ativa → subject (passive)
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