Math 1 (Grade 10)
A geometry and reasoning book: logic and conditionals, parallel lines and slope, congruent triangles and triangle relationships.
A geometry and reasoning book: logic and conditionals, parallel lines and slope, congruent triangles and triangle relationships.
Units of the book
| Unit | What to know |
|---|---|
| Reasoning and proof | inductive reasoning, logic, conditionals and their converse, inverse and contrapositive, algebraic proof |
| Parallel and perpendicular lines | angles formed by a transversal, slope, forms of a line, distance between lines |
| Congruent triangles | classifying triangles, SSS, SAS, ASA, AAS, isosceles triangles |
| Relationships in triangles | bisectors, medians, altitudes, the triangle inequality, indirect proof |
Formulas to remember
- m = (y₂ − y₁)/(x₂ − x₁): slope
- y = mx + b: slope-intercept form
- m₁ · m₂ = −1: perpendicular lines
- |a − b|: triangle inequality
- مجموع زوايا المثلث = 180°: angle sum
- مركز الثقل: 2: 1: centroid
🧠 Memory kit: geometry and line formulas
| Formula | In plain words | How to remember it |
|---|---|---|
| m = (y₂ − y₁) / (x₂ − x₁) | Slope = how much you go up ÷ how much you go across. | y is on top because y is height: “up goes on top”. |
| y − y₁ = m(x − x₁) | The line through a known point with a known slope. | It is the slope formula with both sides multiplied by (x − x₁). |
| y = mx + b | m is the slope, b is where the line crosses the y-axis. | b = “beginning”: the value of y when x = 0. |
| m₁ = m₂ ‖ m₁ · m₂ = −1 | Parallel lines share a slope; perpendicular slopes multiply to −1. | Perpendicular: “flip it and change the sign” (2 → −½). |
| d = √((x₂ − x₁)² + (y₂ − y₁)²) | Distance between two points. | It is Pythagoras: the across and up differences are the legs, the distance is the hypotenuse. |
| M = ((x₁ + x₂)/2, (y₁ + y₂)/2) | The midpoint. | Midpoint = the average of the x’s and the average of the y’s. |
| a² + b² = c² | In a right triangle the square on the hypotenuse equals the two other squares. | Learn the triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25 and their multiples (6-8-10). |
| 30°-60°-90°: x, x√3, 2x | The side opposite 30° is half the hypotenuse. | Small → ×√3 → ×2: “1, root 3, 2”. |
| 45°-45°-90°: x, x, x√2 | Two equal legs; hypotenuse = leg × √2. | Half a square: the hypotenuse is the square’s diagonal. |
| |a − b| | The third side is bigger than the difference and smaller than the sum. | If it equalled the sum, the triangle would flatten into a line. |
| ∠ خارجية = ∠ + ∠ (البعيدتان) | An exterior angle = the sum of the two remote interior angles. | Because the exterior and its neighbour make 180°, exactly like the three angles. |
| (n − 2) × 180° | Angle sum of an n-sided polygon. | From one vertex the polygon splits into (n − 2) triangles of 180° each. |
| 360° / n | Each exterior angle of a regular polygon (they always total 360°). | Walking once around the polygon is one full turn: 360°. |
| n(n − 3) / 2 | Number of diagonals. | Each vertex joins n − 3 others (not itself or its neighbours); halve because each diagonal is counted twice. |
| مركز الثقل 2: 1 | The centroid cuts each median 2: 1 from the vertex. | The long part is always on the vertex side. |
| القطعة المنصفة = ½ الضلع الثالث | The midsegment is parallel to the third side and half its length. | “Middle to middle = half”. |
| k, k², k³ | Similar figures: sides and perimeters ×k, areas ×k², volumes ×k³. | The power = the number of dimensions: length 1, area 2, volume 3. |
| C = 2πr, A = πr² | Circumference and area of a circle. | Area has the “square” because area is always in square units. |
| L = (θ/360°) × 2πr | Arc length = the fraction of the circumference that the angle covers. | A sector works the same way: (θ/360°) × πr². |
| ∠ محيطية = ½ ∠ مركزية | An inscribed angle is half the central angle on the same arc. | So an angle in a semicircle = ½ × 180° = 90°. |
🧠 Memory kit: quadrilaterals, transformations and logic
| Topic | What to know |
|---|---|
| Parallelogram | opposite sides equal and parallel, opposite angles equal, neighbouring angles add to 180°, the diagonals bisect each other. |
| Rectangle | a parallelogram with right angles; its diagonals are equal. |
| Rhombus | a parallelogram with equal sides; its diagonals are perpendicular and bisect the angles. Area = ½ d₁d₂. |
| Square | both a rectangle and a rhombus: every property above. |
| Trapezoid | exactly one pair of parallel sides. Midsegment = half the sum of the bases. Area = ½(a + b)h. |
| Reflection | in the x-axis: (x, −y), in the y-axis: (−x, y), in y = x: (y, x), in the origin: (−x, −y). |
| Rotation about the origin (anticlockwise) | 90°: (−y, x), 180°: (−x, −y), 270°: (y, −x). Trick for 90°: “swap, then change the sign of the first”. |
| Translation and dilation | translation: (x + a, y + b). Dilation by k: (kx, ky); k > 1 enlarges, 0 < k < 1 shrinks. |
| Truth table: “and” | p ∧ q is true only when both are true. |
| Truth table: “or” | p ∨ q is false only when both are false. |
| Conditional | p → q is false in one case only: a true start and a false result (a broken promise). |
| Converse, inverse, contrapositive | converse q → p, inverse ~p → ~q, contrapositive ~q → ~p, which always matches the original; the converse matches the inverse. |
- The broken promise: “if you pass, I’ll give you a prize” is a lie only if you pass and get no prize. That is the only false case of a conditional.
- The contrapositive is the twin of the original: swap and negate together and the truth value never changes.
- Congruence tests: SSS, SAS, ASA, AAS, and HL for right triangles. AAA gives only similarity, and SSA never works.
- Triangle centres: perpendicular bisectors → circumcentre (circle through the vertices), angle bisectors → incentre (circle touching the sides), medians → centroid, altitudes → orthocentre.
A way of memorizing that sticks
- Understand before you memorize: read “in plain words” for each formula and say out loud what it measures and why each quantity is multiplied or divided. A formula you understand takes a minute to learn.
- Active recall: cover the formula column and write it from memory, then check. Writing from memory beats rereading by far.
- Spaced repetition: review the same sheet after 1 day, 3 days, a week and two weeks. Four short reviews lock it in until test day.
- Your own formula page: write every formula of a book on one page by hand. If it does not fit, you are writing things you can derive, so cut them.
- Teach it: explain the formula to a friend, or to yourself in a mirror, in two sentences. If you cannot explain it, you do not know it yet.
- Use it at once: after each group, press “Try one” in the patterns below. Memory sets when you use it.
Question patterns
Patterns
Look for a rule: constant difference, constant ratio, squares, or adding the two previous terms.
Conditional statements
Converse: q → p. Inverse: ~p → ~q. Contrapositive: ~q → ~p, equivalent to the original.
Parallel-line angles
Corresponding and alternate angles are equal; same-side interior angles add to 180°.
Slope, line equations, distance
Parallel lines: equal slopes; perpendicular: product −1. The distance between y = a and y = b is |a − b|.
Triangle angles
Sum 180°; exterior angle = sum of the remote interior angles; isosceles base angles are equal.
Congruence and triangle centres
SSS, SAS, ASA, AAS prove congruence; AAA and SSA do not. Perpendicular bisectors → circumcenter, angle bisectors → incenter, medians → centroid, altitudes → orthocenter.
Triangle inequality and the centroid
The third side is between the difference and the sum. The centroid is ⅔ of the median from the vertex.
- Mixing up converse (swap) and inverse (negate).
- Perpendicular slopes multiply to −1, not 1.
- SSA is not a congruence test.
More patterns (figures and calculations)
Parallel lines with expressions
Name the angle pair: corresponding or alternate → equal; consecutive interior → sum 180°. Then solve.
Triangle angles and the exterior angle
Interior angles add to 180°; an exterior angle equals the two remote interior angles.
Isosceles triangles
The base angles are equal.
Similar triangles
Corresponding sides are proportional: divide matching sides to get the scale factor.
Pythagoras
Know the triples 3-4-5, 5-12-13, 8-15-17, 7-24-25 and their multiples.
Inscribed and central angles
An inscribed angle is half the central angle on the same arc.
Parallelograms
Opposite sides are equal; consecutive angles are supplementary.
The midsegment
It is half the third side and parallel to it.
Polygon angles
(n − 2) × 180 for the interior sum; 360 ÷ n for each exterior angle of a regular polygon.
Transformations
Translation: add. Reflect in x-axis: negate y. Rotate 90°: (x, y) → (−y, x). Rotate 180°: (−x, −y).
Distance and midpoint
d = √((x₂ − x₁)² + (y₂ − y₁)²) and M = ((x₁ + x₂)/2, (y₁ + y₂)/2)