U6 Introduction to Geometry: Essential Formulas

This section covers fundamental geometric concepts, including properties of angles, triangles, polygons, and basic coordinate geometry. Mastery of these formulas is crucial for solving DSE geometry problems.

1 Angles and Parallel Lines

Angles at a Point & on a Straight Line

The sum of angles at a point is $360^\circ$. The sum of angles on a straight line is $180^\circ$.

$$ \angle a + \angle b + \angle c = 360^\circ $$
abc

Vertically Opposite Angles & Angles with Parallel Lines

Vertically opposite angles are equal. For two parallel lines cut by a transversal: corresponding angles are equal, alternate angles are equal, and interior angles are supplementary (sum to $180^\circ$).

$$ \angle a = \angle c, \quad \angle b = \angle d $$

2 Triangles

Angle Sum of a Triangle

The sum of the interior angles in any triangle is $180^\circ$.

$$ \angle A + \angle B + \angle C = 180^\circ $$

Exterior Angle of a Triangle

An exterior angle of a triangle is equal to the sum of the two opposite interior angles.

$$ \angle d = \angle a + \angle b $$

Pythagoras' Theorem

For a right-angled triangle, the square of the length of the hypotenuse $c$ is equal to the sum of the squares of the lengths of the other two sides $a$ and $b$.

$$ a^2 + b^2 = c^2 $$

3 Polygons

Sum of Interior Angles

For an $n$-sided polygon, the sum of its interior angles is $(n-2) \times 180^\circ$.

$$ S_{\text{int}} = (n-2) \times 180^\circ $$

Sum of Exterior Angles

The sum of the exterior angles (one at each vertex) of any convex polygon is $360^\circ$.

$$ S_{\text{ext}} = 360^\circ $$

4 Basic Coordinate Geometry

Distance Formula

The distance $d$ between two points $A(x_1, y_1)$ and $B(x_2, y_2)$ in the Cartesian plane.

$$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

Midpoint Formula

The coordinates of the midpoint $M$ of the line segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$.

$$ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) $$

Slope (Gradient) of a Line

The slope $m$ of a non-vertical line passing through points $A(x_1, y_1)$ and $B(x_2, y_2)$, where $x_1 \neq x_2$.

$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$

Struggling with complex problems?

Learner App features AI step-by-step analysis technology. Snap a photo and it will guide you through the solution!

Download Learner Now