USTET Mathematics — Geometry — Lines, Angles, Polygons, Triangles & CirclesRevision Notes
Final-week revision notes for Geometry — Lines, Angles, Polygons, Triangles & Circles. If you have already studied the full chapter, this page is your go-to refresher before sitting the USTET. Compact, high-yield, and aligned with what University of Santo Tomas tests in the Mathematics subtest.
Exam context
On the USTET 2026, the Mathematics subtest carries a "Core section" weight in University of Santo Tomas's pattern. Geometry — Lines, Angles, Polygons, Triangles & Circles lands at position 5th out of 9 in the standard review order. Target score is Competitive overall score, and roughly a meaningful share of items come from Mathematics on a typical USTET paper.
Geometry — Lines, Angles, Polygons, Triangles & Circles - Revision notes
Geometry forms the foundation of spatial reasoning and is heavily tested in college entrance exams. This chapter covers essential geometric concepts including lines, angles, polygons, triangles, and circles. Master these concepts through problem-solving, formula application, and visual understanding to excel in UPCAT, ACET, USTET, and other entrance examinations.
Sections
Formulas
Example
If line 1 has slope 3/4, then any line parallel to it also has slope 3/4
Formula
m₁ = m₂ (parallel lines)
Variables
m₁, m₂ = slopes of parallel lines
Application
Used to determine if lines are parallel or to find equations of parallel lines
Example
If line 1 has slope 2/3, then a perpendicular line has slope -3/2
Formula
m₁ × m₂ = -1 (perpendicular lines)
Variables
m₁, m₂ = slopes of perpendicular lines
Application
Used to determine if lines are perpendicular or find perpendicular line equations
Example
If one angle is 65°, the adjacent angle is 180° - 65° = 115°
Formula
Linear pair: ∠A + ∠B = 180°
Variables
∠A, ∠B = adjacent angles forming a straight line
Application
Finding unknown angles when two angles form a linear pair
Exam Tips
- Draw clear diagrams and label all angles systematically
- Look for parallel line indicators (arrow marks) in diagrams
- Remember: corresponding and alternate angles are EQUAL, consecutive interior angles are SUPPLEMENTARY
- Practice identifying angle relationships quickly in complex figures
Key Points
- Lines can be parallel, perpendicular, intersecting, or skew depending on their relationship
- When parallel lines are cut by a transversal, several angle relationships are formed
- Corresponding angles are equal when parallel lines are cut by a transversal
- Alternate interior and exterior angles are equal in parallel line configurations
- Consecutive interior angles are supplementary (sum to 180°)
- Vertical angles are always equal when two lines intersect
Definitions
Term
Transversal
Definition
A line that intersects two or more coplanar lines at different points
Importance
Essential for understanding angle relationships in parallel line configurations
Term
Corresponding Angles
Definition
Angles that occupy the same relative position at each intersection where a transversal crosses two lines
Importance
Key to proving lines are parallel and solving angle problems
Term
Alternate Interior Angles
Definition
Non-adjacent angles on opposite sides of a transversal, inside the parallel lines
Importance
Another method to prove parallel lines and solve for unknown angles
Section Title
Lines and Angles
Common Mistakes
- Confusing corresponding angles with alternate angles
- Forgetting that consecutive interior angles are supplementary, not equal
- Mixing up the formulas for parallel vs perpendicular line slopes
- Not identifying the transversal correctly in complex diagrams
Formulas
Example
A hexagon (n=6) has 6(6-3)/2 = 9 diagonals
Formula
Number of diagonals = n(n-3)/2
Variables
n = number of sides
Application
Finding total diagonals in any polygon
Example
Pentagon: (5-2) × 180° = 540°
Formula
Sum of interior angles = (n-2) × 180°
Variables
n = number of sides
Application
Finding total of all interior angles
Example
Regular octagon: (8-2) × 180° ÷ 8 = 135°
Formula
Each interior angle (regular) = (n-2) × 180° ÷ n
Variables
n = number of sides
Application
Finding each angle in a regular polygon
Exam Tips
- Memorize common polygon names: triangle (3), quadrilateral (4), pentagon (5), hexagon (6), etc.
- For regular polygons, use the shortcut: each exterior angle = 360°/n
- Check your work: interior angle + exterior angle = 180°
- Draw the polygon when possible to visualize the problem
Key Points
- A polygon with n sides has n vertices, n angles, and n(n-3)/2 diagonals
- Sum of interior angles of any n-sided polygon = (n-2) × 180°
- Sum of exterior angles of any polygon is always 360°
- Regular polygons have all sides equal and all angles equal
- Each interior angle of a regular n-gon = (n-2) × 180° ÷ n
- Each exterior angle of a regular n-gon = 360° ÷ n
Definitions
Term
Regular Polygon
Definition
A convex polygon where all sides are equal and all interior angles are equal
Importance
Simplifies calculations and appears frequently in exam problems
Term
Diagonal
Definition
A line segment connecting two non-adjacent vertices of a polygon
Importance
Used in polygon classification and area calculations
Term
Exterior Angle
Definition
The angle formed between one side of a polygon and the extension of an adjacent side
Importance
Key property: exterior angles always sum to 360° regardless of polygon type
Section Title
Polygons
Common Mistakes
- Forgetting to subtract 2 in the interior angle sum formula
- Confusing interior and exterior angle relationships
- Miscounting sides or vertices in irregular polygons
- Not recognizing that exterior angle sum is ALWAYS 360°
Formulas
Example
If legs are 3 and 4, then hypotenuse = √(3² + 4²) = 5
Formula
Pythagorean Theorem: a² + b² = c²
Variables
a, b = legs; c = hypotenuse
Application
Finding unknown side length in right triangles
Example
If shortest side = 6, then sides are 6, 6√3, and 12
Formula
30-60-90 triangle ratio = 1 : √3 : 2
Variables
shortest side : medium side : hypotenuse
Application
Quickly finding all sides when one side is known
Example
If leg = 5, then hypotenuse = 5√2
Formula
45-45-90 triangle ratio = 1 : 1 : √2
Variables
leg : leg : hypotenuse
Application
Finding sides in isosceles right triangles
Example
Triangle with base 8 and height 6 has area = ½ × 8 × 6 = 24
Formula
Area of triangle = ½ × base × height
Variables
base = any side; height = perpendicular distance to that side
Application
Finding area when base and height are known
Exam Tips
- Always check if a triangle is right-angled before using Pythagorean theorem
- Memorize special triangle ratios - they appear frequently in exams
- In similar triangles, set up proportions correctly: corresponding sides over corresponding sides
- Use triangle inequality as a quick check for valid triangles
Key Points
- Sum of interior angles in any triangle is always 180°
- Triangle inequality: sum of any two sides > third side
- In a triangle, the largest angle is opposite the longest side
- Triangles can be classified by sides (scalene, isosceles, equilateral) or angles (acute, right, obtuse)
- Pythagorean theorem applies to right triangles: a² + b² = c²
- Special right triangles: 30-60-90 and 45-45-90 have specific side ratios
Definitions
Term
Congruent Triangles
Definition
Triangles that have the same size and shape, with corresponding sides and angles equal
Importance
Proven using SAS, ASA, SSS, or AAS postulates
Term
Similar Triangles
Definition
Triangles that have the same shape but not necessarily the same size, with corresponding angles equal
Importance
Corresponding sides are proportional, useful for solving unknown lengths
Term
Hypotenuse
Definition
The longest side of a right triangle, opposite the right angle
Importance
Central to Pythagorean theorem and right triangle calculations
Section Title
Triangles
Common Mistakes
- Applying Pythagorean theorem to non-right triangles
- Confusing the ratios for 30-60-90 and 45-45-90 triangles
- Forgetting to check triangle inequality when given three side lengths
- Mixing up corresponding parts in similar triangles
Formulas
Example
Circle with radius 7 has circumference = 2π(7) = 14π
Formula
Circumference = 2πr = πd
Variables
r = radius; d = diameter
Application
Finding the distance around a circle
Example
Circle with radius 5 has area = π(5²) = 25π
Formula
Area = πr²
Variables
r = radius
Application
Finding the space inside a circle
Example
If intercepted arc = 80°, then inscribed angle = 40°
Formula
Inscribed angle = ½ × intercepted arc
Variables
inscribed angle = angle with vertex on circle; intercepted arc = arc cut off by the angle
Application
Finding angles or arcs in circle problems
Example
If PA = 6, PB = 4, PC = 8, then PD = (6×4)/8 = 3
Formula
Power of a Point: PA × PB = PC × PD
Variables
P = external point; A,B,C,D = intersection points with circle
Application
Finding unknown lengths in secant-chord problems
Exam Tips
- Remember: central angle = arc measure, inscribed angle = ½ arc measure
- When you see a tangent, immediately think 'perpendicular to radius'
- Use π ≈ 3.14 for numerical calculations unless exact form is required
- Draw radii to connect center with important points on the circle
Key Points
- All radii of a circle are equal; diameter = 2 × radius
- Circumference = 2πr = πd; Area = πr²
- Central angle equals its intercepted arc measure
- Inscribed angle = ½ × intercepted arc measure
- Tangent line is perpendicular to radius at point of tangency
- Angles formed by chords, secants, and tangents have specific relationships
Definitions
Term
Central Angle
Definition
An angle whose vertex is at the center of the circle
Importance
Equals the measure of its intercepted arc
Term
Inscribed Angle
Definition
An angle whose vertex lies on the circle and whose sides pass through two other points on the circle
Importance
Always equals half the intercepted arc measure
Term
Tangent
Definition
A line that intersects a circle at exactly one point
Importance
Always perpendicular to the radius at the point of tangency
Section Title
Circles
Common Mistakes
- Confusing central angles with inscribed angles
- Forgetting that inscribed angles are HALF the intercepted arc
- Not recognizing perpendicular relationship between tangent and radius
- Mixing up formulas for circumference and area
Formulas
Example
Room 12m × 8m has area = 96 m²
Formula
Rectangle Area = length × width
Variables
length, width = perpendicular dimensions
Application
Finding area of rectangular spaces, rooms, fields
Example
Triangle with base 10 and height 7 has area = 35
Formula
Triangle Area = ½ × base × height
Variables
base = any side; height = perpendicular distance to base
Application
Finding area when base and height are given
Example
Trapezoid with bases 6, 10 and height 4 has area = ½(16)(4) = 32
Formula
Trapezoid Area = ½(b₁ + b₂)h
Variables
b₁, b₂ = parallel sides; h = height
Application
Finding area of trapezoid shapes
Exam Tips
- Always identify what you're looking for: perimeter (around) or area (inside)
- Check units - perimeter is linear (m), area is square (m²)
- For composite shapes, break them into familiar shapes
- Double-check that height is perpendicular to the chosen base
Key Points
- Perimeter measures the distance around a shape's boundary
- Area measures the space enclosed within a shape
- Rectangle: P = 2(l + w), A = lw
- Square: P = 4s, A = s²
- Triangle: A = ½bh (base × height)
- Circle: P = 2πr, A = πr²
Definitions
Term
Perimeter
Definition
The total distance around the boundary of a two-dimensional shape
Importance
Used for fencing, borders, and boundary measurements
Term
Area
Definition
The amount of space contained within the boundary of a two-dimensional shape
Importance
Used for flooring, painting, and space calculations
Term
Base and Height
Definition
Base is a chosen side; height is the perpendicular distance from base to opposite vertex/side
Importance
Critical for triangle and parallelogram area calculations
Section Title
Area and Perimeter Formulas
Common Mistakes
- Using slant height instead of perpendicular height
- Forgetting the ½ factor in triangle area formula
- Confusing perimeter and area units (linear vs. square)
- Not converting units before calculating
Connections
- Coordinate geometry builds on line and angle concepts for graphing linear equations
- Trigonometry extends triangle relationships using sine, cosine, and tangent ratios
- Area and volume calculations connect to real-world applications in architecture and engineering
- Circle properties are fundamental to understanding periodic functions and waves in physics
- Polygon angle relationships appear in crystallography and molecular geometry in chemistry
Exam Strategy
Focus on memorizing key formulas and their applications. Practice identifying geometric relationships quickly in diagrams. Work through problems step-by-step, showing all calculations. For multiple choice questions, use elimination by checking if answers satisfy basic geometric principles. Always draw diagrams when not provided - visual representation helps avoid errors and reveals relationships. Time management is crucial: spend more time on multi-step problems that carry higher points.
Quick Review Questions
What is the sum of interior angles in a hexagon?
Using formula (n-2) × 180° where n=6: (6-2) × 180° = 4 × 180° = 720°
If two parallel lines are cut by a transversal and one corresponding angle is 65°, what is the measure of all other corresponding angles?
Corresponding angles are equal when parallel lines are cut by a transversal
In a right triangle with legs 5 and 12, what is the length of the hypotenuse?
Using Pythagorean theorem: c² = 5² + 12² = 25 + 144 = 169, so c = 13
What is the area of a circle with diameter 10?
Diameter = 10, so radius = 5. Area = πr² = π(5²) = 25π
If an inscribed angle intercepts an arc of 120°, what is the measure of the inscribed angle?
Inscribed angle = ½ × intercepted arc = ½ × 120° = 60°
How many diagonals does an octagon have?
Using formula n(n-3)/2 where n=8: 8(8-3)/2 = 8(5)/2 = 20
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Perimeter, Area, Volume & Equation of a Line
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