Geometry Flashcards
Lines and Segments
Line → A straight path that goes on forever in both directions.
Line segment → A part of a line that has two endpoints (it stops at both ends).
Congruent segments → Line segments that are equal in length.
Midpoint → A point that divides a segment exactly in half (creates two equal parts).
Angles & Intersecting Lines
When two lines cross, they create four angles.
Each angle has a vertex at the point of intersection of the two lines.
Vertical (opposite) angles → The angles across from each other when two lines intersect.
These are always equal.
Sum of all four angles at the intersection = 360°.
congruent angles → any angles that have the same measure. It doesn’t matter where they are or how they’re positioned
Types of angles
Acute angle → Less than 90°.
Right angle → Exactly 90°.
Obtuse angle → Between 90° and 180°.
Types of lines
Straight line → A line that extends forever in both directions.
Line segment → A part of a line with two endpoints.
Perpendicular lines (⊥) → Two lines that meet at 90° (a right angle).
Parallel lines (∥) → Two lines that never intersect.
Transversal → A line that crosses two or more other lines.
📐 Polygons
A polygon is a closed shape made of three or more line segments in the same plane.
Sides: The line segments of a polygon.
Vertices: The points where two sides meet.
Convex polygon: A polygon where all interior angles are less than 180°.
🔺 Interior Angle Sum Formula:
For an n-sided polygon:
(n−2)×180°
Examples:
- Triangle (3 sides): 180°
- Quadrilateral (4 sides): 360°
- Pentagon (5 sides): 540°
⭐ Regular Polygon → All sides & angles are congruent.
📏 Perimeter → Sum of all sides.
📦 Area → Space enclosed inside the polygon.
Triangles
Every triangle has three sides and three interior angles.
The measures of the interior angles add up to 180°.
The length of each side must be less than the sum of the lengths of the other two sides.
The following are 3 types of special triangles:
- Equilateral triangle: A triangle with three congruent sides. The measures of the three interior angles of such a triangle are equal, and each measure 60°.
- Isosceles triangle: A triangle with at least two congruent sides. If a triangle has two congruent sides, then the angles opposite the two congruent sides are congruent.
- Right triangle: A triangle with an interior right angle. The side opposite the right angle is called the hypotenuse; the other two sides are called legs.
Area of triangle
The area A of a triangle is given by the formula:
𝐴=𝑏ℎ/2
b: base
h: height
Any side of a triangle can be used as a base; the height that corresponds to the base is the perpendicular line segment from the opposite vertex to the base (or an extension of the base). Depending on the context, the term “base” can also refer to the length of a side of the triangle, and the term “height” can refer to the length of the perpendicular line segment from the opposite vertex to that side.
The Pythagorean Theorem
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.
formula C^2 = A^2 + B^2
C is the hypothenuse (the line opposite to the 90 angle)
A the height line
B the length line
to find A^2 = C^2 - B^2
to find B^2 = C^2 - A^2
Special types of qudrilaterals
Every quadrilateral has four sides and four interior angles. The measures of the interior angles add up to 360°.
The following are four special types of quadrilaterals.
- Rectangle.
- Square
- parallelogram: A quadrilateral in which both pairs of opposite sides are parallel is called a parallelogram. In a parallelogram, opposite sides are congruent and opposite angles are congruent.
- trapezoid: A quadrilateral in which at least one pair of opposite sides is parallel. Two opposite, parallel sides of the trapezoid are called bases of the trapezoid.
Area of parallelograms (square & rectangle)
parallelogram: A quadrilateral in which both pairs of opposite sides are parallel is called a parallelogram. In a parallelogram, opposite sides are congruent and opposite angles are congruent.
For all parallelograms, including rectangles and squares, the area A is given by the formula
𝐴=𝑏ℎ
Area of trapezoid
trapezoid: A quadrilateral in which at least one pair of opposite sides is parallel. Two opposite, parallel sides of the trapezoid are called bases of the trapezoid.
The area A of a trapezoid is given by the formula
𝐴=1/2 (𝑏1+𝑏2) (ℎ)
Circles
The point O is called the center of the circle.
Radius (r): distance from center of the circle to edges.
The diameter of the circle is twice the radius.
Area of the circle
𝜋𝑟2
Circle circumference
- circumference: The distance around a circle.
𝐶=2𝜋𝑟
Arc of the circle
an arc is the part of the circle containing the two points and all the points between them.
To find the length of an arc, use this ratio:
ArcLength / Circumference = Arc’sAngle / 360∘
This means the arc’s length is a fraction of the total circumference, based on its angle.
Area of the circle
A=π r^2
Sector or a Circle
A sector of a circle is a region bounded by an arc of the circle and two radii.
The formula for the area of a sector of a circle is:
SectorArea (S) = (central angle of the sector / 360) ( πr^2)
A tangent to a circle
A tangent is a line that touches a circle at only one point. This special point is called the point of tangency.
📌 Important Rule:
If a line is tangent to a circle, then the radius drawn to the point of tangency is perpendicular (90°) to the tangent line.
🔄 Converse Rule:
If a radius meets a line at a point on the circle and forms a 90° angle, then that line is a tangent
If a question mentions a tangent, immediately check if a 90° angle is involved (helpful for finding missing lengths using the Pythagorean theorem).
Inscribed Polygons
A polygon is inscribed in a circle when all of its vertices lie on the circle. This means the circle perfectly surrounds the polygon.
A polygon is circumscribed around a circle when each of its sides is tangent to the circle. This means the circle is perfectly inside the polygon, touching each side exactly once.
Concentric circles are two or more circles that share the same center but have different radii, like ripples in water.
Three-Dimensional Figures
Basic three-dimensional figures include rectangular solids, cubes, cylinders, spheres, pyramids, and cones. n this section, we look at some properties of rectangular solids and right circular cylinders.
Rectangular Solids (aka Boxes)
Has 6 rectangular faces, 12 edges, and 8 vertices.
Adjacent faces are perpendicular to each other.
The dimensions are length (𝓁), width (𝑤), and height (ℎ).
📏 Formula Cheat Sheet:
Volume (V) = length × width × height → 📦 how much space is inside
Surface Area (A) = add up all the sides → 📏 total area covering the outside.
If a problem involves cutting or reshaping a rectangular solid, focus on how volume stays constant but surface area changes.
Cube
A cube is a special type of rectangular solid where:
ℓ=w=h
Volume: V=s ^3
(Since all sides are equal, cube the side length.)
Surface Area: A=6s ^2
(Since there are 6 equal square faces.)
Cubes often appear in data interpretation questions where you count the number of smaller cubes inside a larger cube.
Right Circular Cylinders
Key Properties
Has two parallel circular bases and a curved lateral surface.
The height (h) is the perpendicular distance between the bases.
The radius (r) is the distance from the center to the edge of the base.
Volume: V=π r^2 h
Surface Area: A=2 π r^2 + 2 π r h
If a problem gives diameter instead of radius, divide by 2 before using formulas.
Be comfortable with factoring out π when simplifying answers.
