USTET Mathematics — Geometry — Lines, Angles, Polygons, Triangles & CirclesCheat Sheet
A printable cheat sheet for Geometry — Lines, Angles, Polygons, Triangles & Circles, built for USTET reviewers who want one go-to reference in the final stretch. Covers formulas, key definitions, common question types, and the University of Santo Tomas-specific twists you will see on USTET day.
Exam context
The University of Santo Tomas Entrance Test is conducted by University of Santo Tomas and is scheduled for Early Q4 2026. The Mathematics subtest is marked as "Core section" in the official pattern, and Geometry — Lines, Angles, Polygons, Triangles & Circles appears in position 5th of 9 in the USTET Mathematics review rotation. Passing mark: Competitive overall score. Recent USTET 2026 papers have drawn roughly a meaningful share of questions from this subject.
Geometry — Lines, Angles, Polygons, Triangles & Circles - Cheat sheet
Your last-minute revision companion for geometry fundamentals. Every formula, theorem, and key concept you need to ace UPCAT and other entrance exams.
Sections
Formulas
Formula
Distance = |x₂ - x₁| (horizontal) or |y₂ - y₁| (vertical)
Meaning
Distance between two points on coordinate plane
Watch Out
Always use absolute value - distance is never negative
When To Use
Finding length of horizontal or vertical line segments
Formula
m = (y₂ - y₁)/(x₂ - x₁)
Meaning
m = slope, (x₁,y₁) and (x₂,y₂) are two points
Watch Out
Undefined slope when x₂ = x₁ (vertical line)
When To Use
Finding slope of line through two points
Formula
y - y₁ = m(x - x₁)
Meaning
Point-slope form: m = slope, (x₁,y₁) = known point
Watch Out
Don't confuse with slope-intercept form y = mx + b
When To Use
Writing equation when you have slope and one point
Section Title
Lines and Angles
Important Facts
- Sum of angles on a straight line = 180°
- Vertically opposite angles are equal
- Corresponding angles are equal when lines are parallel
- Alternate interior angles are equal when lines are parallel
- Consecutive interior angles are supplementary (180°) when lines are parallel
Key Definitions
Term
Parallel Lines
Example
m₁ = m₂ = 2 for lines y = 2x + 1 and y = 2x + 5
Definition
Lines with equal slopes that never intersect
Term
Perpendicular Lines
Example
If m₁ = 2, then m₂ = -1/2
Definition
Lines whose slopes are negative reciprocals
Term
Vertical Angles
Example
Always equal: ∠1 = ∠3, ∠2 = ∠4
Definition
Opposite angles formed by two intersecting lines
Diagrams To Know
- Parallel lines cut by transversal showing all 8 angles
- Perpendicular lines forming 90° angles
- Coordinate plane with positive/negative quadrants
Formulas
Formula
Area = (1/2) × base × height
Meaning
Base = any side, height = perpendicular distance to that side
Watch Out
Height must be perpendicular to the base
When To Use
Finding triangle area with base and height given
Formula
Perimeter = a + b + c
Meaning
a, b, c = lengths of the three sides
Watch Out
All three sides must be given
When To Use
Finding total distance around triangle
Formula
a² + b² = c²
Meaning
Pythagorean theorem: a, b = legs, c = hypotenuse
Watch Out
Only works for right triangles (90° angle)
When To Use
Right triangles only - finding unknown side
Formula
Sum of interior angles = 180°
Meaning
∠A + ∠B + ∠C = 180°
Watch Out
Works for all triangles, not just special ones
When To Use
Finding unknown angle in any triangle
Common Values
Value
1 : √3 : 2
Symbol
short : medium : long
Quantity
30-60-90 triangle sides
Value
1 : 1 : √2
Symbol
leg : leg : hypotenuse
Quantity
45-45-90 triangle sides
Value
3-4-5, 5-12-13, 8-15-17
Symbol
a-b-c
Quantity
Common Pythagorean triples
Section Title
Triangles
Important Facts
- Triangle inequality: sum of any two sides > third side
- Exterior angle = sum of two non-adjacent interior angles
- In right triangle: side opposite largest angle is longest
- Similar triangles have proportional sides and equal angles
- Altitude creates two right triangles in original triangle
Key Definitions
Term
Equilateral Triangle
Example
Side = 5, all angles = 60°
Definition
All sides equal, all angles = 60°
Term
Isosceles Triangle
Example
If AB = AC, then ∠B = ∠C
Definition
Two sides equal, two angles equal
Term
Right Triangle
Example
3-4-5 triangle where 3² + 4² = 5²
Definition
One angle = 90°
Term
Congruent Triangles
Example
SSS, SAS, ASA, AAS prove congruence
Definition
Same size and shape - all corresponding parts equal
Diagrams To Know
- 30-60-90 triangle with sides in ratio 1:√3:2
- 45-45-90 triangle with sides in ratio 1:1:√2
- Right triangle showing altitude to hypotenuse
Formulas
Formula
Sum of interior angles = (n-2) × 180°
Meaning
n = number of sides
Watch Out
Formula only works for n ≥ 3
When To Use
Finding total of all interior angles in any polygon
Formula
Each interior angle = (n-2) × 180° ÷ n
Meaning
For regular polygons only (all angles equal)
Watch Out
Only for regular polygons - irregular polygons have different angles
When To Use
Finding one angle in regular polygon
Formula
Number of diagonals = n(n-3)/2
Meaning
n = number of vertices/sides
Watch Out
Don't count sides as diagonals
When To Use
Counting all possible diagonals in polygon
Formula
Sum of exterior angles = 360°
Meaning
Always 360° regardless of number of sides
Watch Out
True for all polygons, not just regular ones
When To Use
Finding exterior angles or checking work
Section Title
Polygons
Important Facts
- Triangle: 3 sides, sum of angles = 180°
- Quadrilateral: 4 sides, sum of angles = 360°
- Pentagon: 5 sides, each interior angle = 108° (if regular)
- Hexagon: 6 sides, each interior angle = 120° (if regular)
- Octagon: 8 sides, each interior angle = 135° (if regular)
Key Definitions
Term
Regular Polygon
Example
Square, equilateral triangle, regular hexagon
Definition
All sides equal and all angles equal
Term
Convex Polygon
Example
Rectangle, pentagon - no sides bend inward
Definition
All interior angles less than 180°
Term
Diagonal
Example
In square ABCD, AC and BD are diagonals
Definition
Line segment connecting non-adjacent vertices
Diagrams To Know
- Regular polygons: triangle through octagon
- Convex vs concave polygon examples
- Polygon with diagonals drawn from one vertex
Formulas
Formula
Area of rectangle = length × width
Meaning
l = length, w = width
Watch Out
Make sure you're using perpendicular sides
When To Use
Finding area of rectangle or square
Formula
Area of parallelogram = base × height
Meaning
Height is perpendicular distance between parallel sides
Watch Out
Height is NOT the side length - it's perpendicular
When To Use
Finding area when you have base and perpendicular height
Formula
Area of trapezoid = (1/2)(b₁ + b₂) × h
Meaning
b₁, b₂ = parallel sides, h = height between them
Watch Out
Height must be perpendicular to both parallel sides
When To Use
Finding area of trapezoid
Formula
Area of rhombus = (1/2) × d₁ × d₂
Meaning
d₁, d₂ = lengths of diagonals
Watch Out
Diagonals of rhombus are perpendicular
When To Use
Finding rhombus area using diagonals
Section Title
Quadrilaterals
Important Facts
- All squares are rectangles and rhombuses
- Diagonals of rectangle are equal and bisect each other
- Diagonals of rhombus are perpendicular and bisect each other
- Opposite angles in parallelogram are equal
- Consecutive angles in parallelogram are supplementary
Key Definitions
Term
Square
Example
4 equal sides, 4 right angles
Definition
Rectangle with all sides equal
Term
Rectangle
Example
Opposite sides equal and parallel
Definition
Parallelogram with 4 right angles
Term
Rhombus
Example
Like a diamond shape, diagonals bisect at right angles
Definition
Parallelogram with all sides equal
Term
Parallelogram
Example
Opposite sides equal, opposite angles equal
Definition
Quadrilateral with opposite sides parallel
Diagrams To Know
- Square with diagonals showing 45° angles
- Rectangle with diagonal creating right triangles
- Parallelogram showing height vs side length
- Trapezoid with height marked perpendicular to bases
Formulas
Formula
Circumference = 2πr = πd
Meaning
r = radius, d = diameter = 2r
Watch Out
Use π ≈ 3.14 or leave answer in terms of π
When To Use
Finding distance around circle
Formula
Area = πr²
Meaning
r = radius
Watch Out
Remember to square the radius, not multiply by 2
When To Use
Finding space inside circle
Formula
Arc length = (θ/360°) × 2πr
Meaning
θ = central angle in degrees, r = radius
Watch Out
Angle must be in degrees for this formula
When To Use
Finding length of part of circle circumference
Formula
Sector area = (θ/360°) × πr²
Meaning
θ = central angle in degrees, r = radius
Watch Out
This is area, not arc length
When To Use
Finding area of pie-slice portion of circle
Formula
Inscribed angle = (1/2) × central angle
Meaning
Angle at circumference is half the central angle
Watch Out
Only works for angles with vertex on the circle
When To Use
Finding angles inscribed in circles
Common Values
Value
≈ 3.14159 or 22/7
Symbol
π
Quantity
π (pi)
Section Title
Circles
Important Facts
- Inscribed angle in semicircle is always 90°
- Tangent is perpendicular to radius at point of tangency
- Equal chords are equidistant from center
- Inscribed angles subtending same arc are equal
- Opposite angles of inscribed quadrilateral sum to 180°
Key Definitions
Term
Radius
Example
If center is O and point A is on circle, OA is radius
Definition
Distance from center to any point on circle
Term
Diameter
Example
Longest possible chord in a circle
Definition
Chord that passes through center, equals 2 × radius
Term
Chord
Example
Diameter is the longest chord
Definition
Line segment with both endpoints on circle
Term
Tangent
Example
Perpendicular to radius at point of contact
Definition
Line that touches circle at exactly one point
Term
Secant
Example
Extended chord becomes secant line
Definition
Line that intersects circle at two points
Diagrams To Know
- Circle with center, radius, diameter, chord labeled
- Tangent line perpendicular to radius
- Inscribed angle vs central angle relationship
- Circle with inscribed triangle/quadrilateral
Must Remember
- Pythagorean theorem: a² + b² = c² (right triangles only)
- Sum of triangle angles = 180°
- Circle area = πr², circumference = 2πr
- Parallel lines have equal slopes, perpendicular lines have negative reciprocal slopes
- Sum of polygon interior angles = (n-2) × 180°
- Inscribed angle = half the central angle
- 30-60-90 triangle sides ratio: 1:√3:2
- 45-45-90 triangle sides ratio: 1:1:√2
- Area formulas: triangle = (1/2)bh, rectangle = lw, trapezoid = (1/2)(b₁+b₂)h
- All exterior angles of any polygon sum to 360°
Last Minute Tips
- For right triangles, always check if it's a special triangle (30-60-90 or 45-45-90) before using Pythagorean theorem
- When finding polygon angles, write (n-2)×180° first, then divide by n for regular polygons
- Circle problems: if radius doubles, area becomes 4 times larger (r² effect)
- For parallel line problems, identify the transversal first, then use corresponding/alternate angle rules
- Triangle inequality check: each side must be less than sum of other two sides
Comparison Tables
Rows
Values
- All equal
- Two equal
- All different
Property
By sides
Values
- All 60°
- Two equal
- All different
Property
By angles
Values
- 5-5-5
- 5-5-7
- 3-4-5
Property
Example sides
Columns
- Classification
- Equilateral
- Isosceles
- Scalene
Table Title
Triangle Types by Sides vs Angles
Rows
Values
- Yes
- No
- Yes
- No
Property
All sides equal
Values
- Yes
- Yes
- No
- No
Property
All angles 90°
Values
- Yes
- Yes
- No
- No
Property
Diagonals equal
Values
- Yes
- No
- Yes
- No
Property
Diagonals perpendicular
Columns
- Property
- Square
- Rectangle
- Rhombus
- Parallelogram
Table Title
Quadrilateral Properties
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