Comprehensive Study Notes on 7th Grade Mathematics: Geometry and Algebra

Course Overview and Administrative Details

This study guide is derived from a Mathematics exercise sheet (Ficha de Matemática) designed for the 7th Grade (7° A/B) at Colégio Elo, specifically for the Janga unit. The material was prepared by Professor Jorge Sales for the 2026 academic year. The curriculum covered in this document spans fundamental Euclidean geometry—including angles, transversals, and triangle properties—as well as introductory algebraic expressions and their evaluation. Students are expected to apply definitions, geometric theorems, and algebraic manipulation to solve theoretical and practical problems.

Classification and Properties of Angles

In the study of geometry, angles are classified based on the measure of their opening. A null angle (ângulo nulo) measures exactly $0^{\circ}$. An acute angle (ângulo agudo) is any angle with a measure strictly greater than $0^{\circ}$ and less than $90^{\circ}$. A right angle (ângulo reto) measures exactly $90^{\circ}$ and is often denoted by a small square with a dot. An obtuse angle (ângulo obtuso) measures greater than $90^{\circ}$ but less than $180^{\circ}$. Finally, a straight angle (ângulo raso) measures exactly $180^{\circ}$.

Angular relationships also involve properties of congruence and summation. Congruent angles (ângulos congruentes) are those that possess the exact same numerical measure, such as the angle $\angle IHG$ and any other angle with an identical opening. Complementary angles (ângulos complementares) are pairs of angles whose sum is exactly $90^{\circ}$. For example, the complement of angle $\angle CBA$ is found by subtracting its measure from $90^{\circ}$. Supplementary angles (ângulos suplementares) are pairs whose sum is exactly $180^{\circ}$. To find the supplementary angle of $\angle ZXV$, one must calculate $180^{\circ} - \text{med}(\angle ZXV)$.

Angle Bisectors

An angle bisector (bissetriz) is a ray that originates at the vertex of an angle and divides it into two congruent parts. If a ray $BD$ is the bisector of an angle $\angle ABC$, it creates two smaller angles, $\angle ABD$ and $\angle CBD$, such that $\text{med}(\angle ABD) = \text{med}(\angle CBD) = \frac{1}{2} \text{med}(\angle ABC)$.

For instance, if $BD$ bisects $\angle ABC$ and the measure of $\angle CBD$ is given as $35^{\circ}$, then the measure of $\angle ABD$ must also be $35^{\circ}$, making the total measure of $\angle ABC$ equal to $70^{\circ}$. Similarly, if a ray $FH$ is the bisector of $\angle EFG$, the measure of the whole angle $\angle EFG$ will always be twice the measure of the half-angle $\angle EFH$, and $\angle EFH$ will be congruent to $\angle GFH$.

Angles Formed by Parallel Lines and Transversals

When two parallel lines are intersected by a transversal line, several pairs of angles are formed with specific names and properties based on their relative positions. Alternate Interior Angles (alternos internos) are on opposite sides of the transversal and inside the parallel lines; they are congruent. Alternate Exterior Angles (alternos externos) are on opposite sides of the transversal and outside the parallel lines; these are also congruent. Corresponding Angles (correspondentes) occupy the same relative position at each intersection where a straight line crosses two others; they are congruent if the lines are parallel. Consecutive (or Co-interior/Co-exterior) Angles are on the same side of the transversal and their sum is $180^{\circ}$.

In the exercises provided, specific pairs such as $D$ and $F$, $A$ and $G$, $E$ and $C$, $H$ and $B$, and $B$ and $F$ must be categorized according to these definitions. For example, if $A$ and $E$ are in the same relative position at different intersections, they are labeled as corresponding angles (correspondentes). If $A$ and $G$ are on opposite sides of the transversal outside the parallel boundaries, they are alternate exterior angles.

Classification and Properties of Triangles

Triangles are classified by both their side lengths and their internal angle measures. Based on sides, an equilateral triangle (equilátero) has three equal sides; an isosceles triangle (isósceles) has at least two equal sides; and a scalene triangle (escaleno) has no equal sides. It is a critical geometric fact that every equilateral triangle is also an isosceles triangle, but not every isosceles triangle is equilateral. Based on angles, an acute-angled triangle (acutângulo) has three acute angles; a right-angled triangle (retângulo) has one right angle; and an obtuse-angled triangle (obtusângulo) has one obtuse angle. Every equilateral triangle is inherently acute-angled, as all its angles measure $60^{\circ}$.

The Triangle Inequality Theorem states that for any triangle, the sum of the lengths of any two sides must be strictly greater than the length of the remaining side ($a + b > c$). Applying this:

1. Sides of $5\,cm, 5\,cm, 5\,cm$: Possible, as $5 + 5 > 5$.
2. Sides of $8\,cm, 4\,cm, 3\,cm$: Impossible, because $4 + 3 = 7$, which is less than $8$.
3. Sides of $6\,cm, 4\,cm, 3\,cm$: Possible, as $4 + 3 > 6$.
4. Sides of $8\,cm, 4\,cm, 4\,cm$: Impossible, because $4 + 4 = 8$, which is not strictly greater than $8$.

Internal Angles of a Triangle

The sum of the internal angles of any triangle in Euclidean geometry is always exactly $180^{\circ}$. This is expressed by the formula $\text{med}(\hat{A}) + \text{med}(\hat{B}) + \text{med}(\hat{C}) = 180^{\circ}$.

In a specific case where $\text{med}(\hat{A}) = 48^{\circ}$ and $\text{med}(\hat{B}) = 98^{\circ}$, the measure of angle $\hat{C}$ is calculated as $180^{\circ} - (48^{\circ} + 98^{\circ}) = 180^{\circ} - 146^{\circ} = 34^{\circ}$.

In a more complex algebraic scenario where the measure of angle $C$ is triple that of angle $B$ ($C = 3B$) and the measure of angle $A$ is double that of angle $B$ ($A = 2B$), we set up the equation $2B + B + 3B = 180^{\circ}$. This simplifies to $6B = 180^{\circ}$, resulting in $B = 30^{\circ}$. Consequently, $A = 2 \times 30^{\circ} = 60^{\circ}$ and $C = 3 \times 30^{\circ} = 90^{\circ}$.

Introduction to Algebraic Expressions

Algebraic expressions use variables (letters) to represent unknown or changing values. In the context of Pedro's body mass ($x$ kilograms), gaining $1.5\,kg$ is represented by the expression $x + 1.5$, whereas losing $3\,kg$ is represented by $x - 3$.

Verbal descriptions can be translated into algebraic notation as follows:

• Double of a number: $2n$
• A number plus two: $n + 2$
• Two minus a number: $2 - n$
• A number minus two: $n - 2$
• Two plus the double of a number: $2 + 2n$
• The double of a number minus two: $2n - 2$

Manipulation and Simplification of Algebraic Expressions

Algebraic expressions are composed of terms. Each term consists of a numerical coefficient (coeficiente) and a literal/variable part (parte literal). Similar terms (termos semelhantes) are those that have identical literal parts.

Consider the expression $8x - 4 - x + 17$. The similar terms are grouped: the variable terms $8x$ and $-x$, and the constant terms $-4$ and $17$.

• To simplify, combine the variable terms: $8x - 1x = 7x$.
• Combine the constant terms: $-4 + 17 = 13$.
• The simplified expression is $7x + 13$.

To find the numerical value (valor numérico) of an expression, substitute the variable with a specific number. For instance, if $x = 3$ in the expression $8x - 4 - x + 17$, the calculation is as follows: $8(3) - 4 - (3) + 17 = 24 - 4 - 3 + 17 = 20 - 3 + 17 = 17 + 17 = 34$. Alternatively, using the simplified form $7x + 13$: $7(3) + 13 = 21 + 13 = 34$.