Skip to main content
Super Tutor
Study NotesUSTET · MathematicsReal content

USTET Mathematics — Geometry — Lines, Angles, Polygons, Triangles & CirclesStudy Notes

SummaryConceptsNotesRevisionDeep DiveSlidesFlashcardsMistakesCheatMnemonicsTemplates

Study notes for Geometry — Lines, Angles, Polygons, Triangles & Circles that match the USTET 2026 syllabus. Built to mirror how University of Santo Tomas structures USTET Mathematics questions, these notes walk through each concept with examples, formulas, and practice questions designed for time-pressured exam conditions.

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 - Study notes

Geometry is the branch of mathematics that deals with shapes, sizes, positions, angles, and dimensions of things. This comprehensive study guide covers the fundamental concepts of geometry that are essential for UPCAT and other college entrance examinations in the Philippines. We'll explore lines, angles, polygons, triangles, and circles through problem-solving approaches with step-by-step solutions to help you master these concepts.

Summary

Geometry is fundamental to mathematics and essential for UPCAT success. Key concepts include: **Points, lines, and planes** form the foundation; **angles** have six types and important relationships (complementary, supplementary, vertical); **parallel lines** with transversals create predictable angle patterns; **polygons** follow formulas for angles and diagonals; **triangles** classify by sides and angles with the crucial 180° angle sum; **congruent and similar triangles** follow specific postulates; **right triangles** use the Pythagorean theorem and special angle relationships; **circles** have extensive properties involving angles, chords, and tangents; **formulas** for perimeter, area, and volume are essential tools. Success requires understanding concepts, memorizing key formulas, and practicing problem-solving techniques. Focus on step-by-step approaches, verify answers, and recognize common problem patterns. These geometry skills form the foundation for advanced mathematics and are heavily tested in Philippine college entrance examinations.

Sections

Before diving into specific shapes, let's establish the basic building blocks of geometry. A **point** indicates position or location with no dimension, represented by a dot and named with capital letters (like point A). A **line** is infinitely long and straight with no thickness, connecting any two points. A **plane** extends without limit in every direction and contains infinite points and lines. **Collinear points** lie on the same line, while **noncollinear points** do not. **Coplanar points** lie in the same plane. These definitions form the foundation for understanding all geometric concepts. **Key Postulates:** 1. Any two points determine exactly one line 2. Any three noncollinear points determine exactly one plane 3. Two lines intersect in exactly one point 4. Distance between two points is the absolute value of the difference between their coordinates 5. **Segment Addition Postulate**: If A, B, and C are collinear with B between A and C, then AB + BC = AC

Heading

Fundamental Concepts of Geometry

Examples

  • **Example 1:** On a line, if AB = 5 cm and BC = 8 cm, find AC. **Solution:** Using the Segment Addition Postulate: AC = AB + BC = 5 + 8 = 13 cm
  • **Example 2:** If AC = 15 cm and AB = 7 cm, find BC. **Solution:** AC = AB + BC → 15 = 7 + BC → BC = 15 - 7 = 8 cm

Key Points

  • Point = position with no dimension
  • Line = infinite straight path with no thickness
  • Plane = infinite flat surface
  • Collinear = points on same line
  • Coplanar = points on same plane
  • Segment Addition: AB + BC = AC when B is between A and C

A **line segment** is composed of two endpoints and all points between them, denoted as AB̅ or B̅A̅. A **ray** starts from an endpoint and extends infinitely in one direction, written as A⃗B⃗ (A is the endpoint). **Midpoint and Bisectors:** A **bisector** intersects a line segment at its midpoint, creating two equal parts. The **Midpoint Postulate** states that every line segment has exactly one midpoint. **Types of Lines:** - **Intersecting lines**: Coplanar lines sharing exactly one point - **Perpendicular lines**: Intersecting lines forming 90° angles (⊥) - **Parallel lines**: Coplanar lines that never intersect (∥) - **Skew lines**: Noncoplanar lines that don't intersect

Heading

Lines and Line Segments

Examples

  • **Example 1:** If P is the midpoint of AB̅ and AP = 2x, PB = x + 4, find AB. **Solution:** Step 1: Since P is midpoint, AP = PB Step 2: 2x = x + 4 Step 3: 2x - x = 4 → x = 4 Step 4: AP = 2(4) = 8, PB = 4 + 4 = 8 Step 5: AB = AP + PB = 8 + 8 = 16 units
  • **Example 2:** Find the midpoint of segment with endpoints (2, 5) and (8, 11). **Solution:** Midpoint formula: ((x₁ + x₂)/2, (y₁ + y₂)/2) = ((2 + 8)/2, (5 + 11)/2) = (5, 8)

Key Points

  • Line segment has two endpoints
  • Ray has one endpoint, extends infinitely
  • Midpoint divides segment into equal parts
  • Perpendicular lines form 90° angles
  • Parallel lines never intersect
  • Skew lines are noncoplanar

An **angle** is formed by two rays with a common endpoint (vertex). We measure angles in degrees using a protractor. **Angle Classifications:** - **Acute angle**: 0° < θ < 90° - **Right angle**: θ = 90° - **Obtuse angle**: 90° < θ < 180° - **Straight angle**: θ = 180° - **Reflex angle**: 180° < θ < 360° - **Round angle**: θ = 360° **Angle Relationships:** - **Complementary angles**: Two angles that sum to 90° - **Supplementary angles**: Two angles that sum to 180° - **Adjacent angles**: Share a common side but no interior points - **Vertical angles**: Opposite angles formed by intersecting lines (always equal) - **Linear pair**: Adjacent angles that form a straight line (supplementary) **Angle Addition Postulate**: If C is in the interior of ∠BAD, then m∠BAD = m∠BAC + m∠CAD

Heading

Angles and Angle Relationships

Examples

  • **Example 1:** An angle is three times its supplement. Find the angle. **Solution:** Step 1: Let x = the angle, then (180 - x) = its supplement Step 2: x = 3(180 - x) Step 3: x = 540 - 3x Step 4: x + 3x = 540 → 4x = 540 Step 5: x = 135° Verification: Supplement = 180 - 135 = 45°; 135 = 3(45) ✓
  • **Example 2:** Find the complement of an angle measuring 38°. **Solution:** Complement = 90° - 38° = 52°
  • **Example 3:** If two angles are supplementary and one is 4 times the other, find both angles. **Solution:** Step 1: Let x = smaller angle, 4x = larger angle Step 2: x + 4x = 180° Step 3: 5x = 180° → x = 36° Step 4: Angles are 36° and 144°

Key Points

  • Angles measured in degrees (0° to 360°)
  • Complementary angles sum to 90°
  • Supplementary angles sum to 180°
  • Vertical angles are equal
  • Linear pairs are supplementary
  • Angle bisector creates two equal angles

When a **transversal** (a line crossing two or more lines) intersects parallel lines, it creates eight angles with special relationships. **Angle Pairs with Parallel Lines:** 1. **Corresponding angles** are equal (same relative position) 2. **Alternate interior angles** are equal (on opposite sides, between the parallel lines) 3. **Alternate exterior angles** are equal (on opposite sides, outside the parallel lines) 4. **Consecutive interior angles** are supplementary (same side, between parallel lines) 5. **Consecutive exterior angles** are supplementary (same side, outside parallel lines) **Important:** These relationships only hold when lines are parallel. Conversely, if any of these relationships exist, the lines must be parallel.

Heading

Parallel Lines and Transversals

Examples

  • **Example 1:** If l₁ ∥ l₂, ∠3 = 3x - 30°, and ∠6 = 2x - 6°, find ∠8. **Solution:** Step 1: ∠3 and ∠6 are alternate interior angles, so they're equal Step 2: 3x - 30 = 2x - 6 Step 3: 3x - 2x = 30 - 6 → x = 24 Step 4: ∠6 = 2(24) - 6 = 42° Step 5: ∠8 and ∠6 are supplementary → ∠8 = 180° - 42° = 138°
  • **Example 2:** Two parallel lines cut by a transversal create corresponding angles of (5x + 10)° and (3x + 50)°. Find x. **Solution:** Step 1: Corresponding angles are equal Step 2: 5x + 10 = 3x + 50 Step 3: 5x - 3x = 50 - 10 Step 4: 2x = 40 → x = 20

Key Points

  • Transversal creates 8 angles with parallel lines
  • Corresponding angles are equal
  • Alternate interior/exterior angles are equal
  • Consecutive interior/exterior angles are supplementary
  • These properties prove lines are parallel

A **polygon** is a closed figure formed by three or more line segments where each segment intersects exactly two others at their endpoints. **Polygon Classifications:** - **Convex polygon**: All interior angles < 180° - **Concave polygon**: At least one interior angle > 180° - **Regular polygon**: All sides and angles are equal - **Irregular polygon**: Sides and/or angles are not all equal **Polygon Names by Sides:** - Triangle (3), Quadrilateral (4), Pentagon (5), Hexagon (6), Heptagon (7), Octagon (8), Nonagon (9), Decagon (10) **Important Formulas:** - **Number of diagonals**: D = n(n-3)/2 - **Sum of interior angles**: S = 180°(n-2) - **Each interior angle of regular polygon**: 180°(n-2)/n - **Sum of exterior angles**: Always 360° for any polygon - **Each exterior angle of regular polygon**: 360°/n

Heading

Polygons

Examples

  • **Example 1:** Find the sum of interior angles of a 12-gon. **Solution:** Using S = 180°(n-2): S₁₂ = 180°(12-2) = 180°(10) = 1800°
  • **Example 2:** Each interior angle of a regular polygon is 140°. How many sides? **Solution:** Step 1: Each interior angle = 180°(n-2)/n Step 2: 140 = 180(n-2)/n Step 3: 140n = 180(n-2) Step 4: 140n = 180n - 360 Step 5: 360 = 180n - 140n = 40n Step 6: n = 9 sides (nonagon)
  • **Example 3:** How many diagonals does an octagon have? **Solution:** Using D = n(n-3)/2: D = 8(8-3)/2 = 8(5)/2 = 20 diagonals

Key Points

  • Polygon = closed figure with n sides
  • Convex = all interior angles < 180°
  • Regular = all sides and angles equal
  • Diagonals formula: n(n-3)/2
  • Interior angles sum: 180°(n-2)
  • Exterior angles always sum to 360°

A **triangle** is a polygon with three sides and three angles. The sum of interior angles in any triangle is always 180°. **Classification by Sides:** - **Scalene**: No equal sides - **Isosceles**: Two equal sides (base angles are equal) - **Equilateral**: All sides equal (all angles = 60°) **Classification by Angles:** - **Acute**: All angles < 90° - **Right**: One angle = 90° - **Obtuse**: One angle > 90° - **Equiangular**: All angles equal (60° each) **Important Elements:** - **Altitude**: Perpendicular from vertex to opposite side - **Median**: Line from vertex to midpoint of opposite side - **Angle bisector**: Divides an angle into two equal parts **Triangle Inequality Theorem**: Sum of any two sides > third side For sides a, b, c: a + b > c, a + c > b, b + c > a

Heading

Triangles: Classification and Properties

Examples

  • **Example 1:** Can sides 5, 12, 8 form a triangle? **Solution:** Check triangle inequality: 5 + 12 = 17 > 8 ✓ 5 + 8 = 13 > 12 ✓ 12 + 8 = 20 > 5 ✓ Yes, they can form a triangle.
  • **Example 2:** In triangle ABC, ∠A = 40°, ∠B = 60°. Find ∠C. **Solution:** ∠A + ∠B + ∠C = 180° 40° + 60° + ∠C = 180° ∠C = 180° - 100° = 80°
  • **Example 3:** An isosceles triangle has a vertex angle of 40°. Find the base angles. **Solution:** Step 1: Let base angles = x (they're equal) Step 2: 40° + x + x = 180° Step 3: 40° + 2x = 180° Step 4: 2x = 140° → x = 70° Base angles are both 70°.

Key Points

  • Sum of angles = 180°
  • Isosceles triangle has equal base angles
  • Equilateral triangle has all 60° angles
  • Triangle inequality: sum of two sides > third side
  • Right triangle has one 90° angle
  • Exterior angle = sum of opposite interior angles

**Congruent triangles** have exactly the same shape and size. All corresponding sides and angles are equal. **Congruence Postulates:** 1. **SSS** (Side-Side-Side): All three sides equal 2. **SAS** (Side-Angle-Side): Two sides and included angle equal 3. **ASA** (Angle-Side-Angle): Two angles and included side equal 4. **SAA** (Side-Angle-Angle): Two angles and non-included side equal **Similar triangles** have the same shape but different sizes. Corresponding angles are equal, and corresponding sides are proportional. **Similarity Postulates:** 1. **AAA** (Angle-Angle-Angle): All corresponding angles equal 2. **SSS** (Side-Side-Side): All corresponding sides proportional 3. **SAS** (Side-Angle-Side): Two sides proportional and included angle equal **Key:** Use ~ for similar (△ABC ~ △DEF) and ≅ for congruent (△ABC ≅ △DEF)

Heading

Congruent and Similar Triangles

Examples

  • **Example 1:** Triangles RST and XYZ are similar with sides in ratio 3:5. If RS = 12, find XY. **Solution:** Step 1: Set up proportion: RS/XY = 3/5 Step 2: 12/XY = 3/5 Step 3: Cross multiply: 3(XY) = 5(12) Step 4: 3(XY) = 60 → XY = 20
  • **Example 2:** In similar triangles, if the ratio of corresponding sides is 2:3, and the smaller triangle's perimeter is 24, find the larger triangle's perimeter. **Solution:** Step 1: Perimeter ratio = side ratio = 2:3 Step 2: 24/P = 2/3 Step 3: 2P = 72 → P = 36

Key Points

  • Congruent = same shape and size
  • Similar = same shape, different size
  • Congruence: SSS, SAS, ASA, SAA
  • Similarity: AAA, SSS, SAS
  • Similar triangles have proportional sides
  • Corresponding angles in similar triangles are equal

A **right triangle** has one 90° angle. The side opposite the right angle is the **hypotenuse** (longest side), while the other two sides are **legs**. **Pythagorean Theorem**: In a right triangle with legs a and b, and hypotenuse c: **a² + b² = c²** **Special Right Triangles:** **45°-45°-90° Triangle:** - Legs are equal: if leg = x, then hypotenuse = x√2 - Ratio of sides = 1 : 1 : √2 **30°-60°-90° Triangle:** - If shortest side = x, then longer leg = x√3, hypotenuse = 2x - Ratio of sides = 1 : √3 : 2 **3-4-5 Triangle:** - Pythagorean triple: 3² + 4² = 5² - Other common triples: 5-12-13, 8-15-17, 7-24-25

Heading

Right Triangles and Pythagorean Theorem

Examples

  • **Example 1:** Find the hypotenuse of a right triangle with legs 9 and 12. **Solution:** Step 1: Use a² + b² = c² Step 2: 9² + 12² = c² Step 3: 81 + 144 = c² Step 4: 225 = c² → c = 15
  • **Example 2:** In a 45°-45°-90° triangle, if the hypotenuse is 10√2, find each leg. **Solution:** Step 1: In 45°-45°-90°, hypotenuse = leg × √2 Step 2: 10√2 = leg × √2 Step 3: leg = 10√2 ÷ √2 = 10
  • **Example 3:** A 30°-60°-90° triangle has shortest side 6. Find the other sides. **Solution:** Step 1: Shortest side = 6 Step 2: Longer leg = 6√3 Step 3: Hypotenuse = 2(6) = 12 Sides are 6, 6√3, and 12.

Key Points

  • Pythagorean theorem: a² + b² = c²
  • Hypotenuse is longest side
  • 45°-45°-90° ratio: 1 : 1 : √2
  • 30°-60°-90° ratio: 1 : √3 : 2
  • Common Pythagorean triples: 3-4-5, 5-12-13
  • Use theorem to find unknown sides

A **circle** is the set of all points equidistant from a fixed point called the **center**. The distance from center to any point on the circle is the **radius**. **Circle Terminology:** - **Radius**: Distance from center to circle (r) - **Diameter**: Chord through center; equals 2r - **Chord**: Line segment with endpoints on circle - **Secant**: Line intersecting circle at two points - **Tangent**: Line touching circle at exactly one point - **Arc**: Portion of circle between two points - **Central angle**: Angle with vertex at center - **Inscribed angle**: Angle with vertex on circle **Important Formulas:** - **Circumference**: C = 2πr = πd - **Area**: A = πr² - **Arc length**: s = (θ/360°) × 2πr (θ in degrees) - **Sector area**: A = (θ/360°) × πr² (θ in degrees)

Heading

Circles: Basic Concepts and Properties

Examples

  • **Example 1:** Find the area and circumference of a circle with radius 7 cm. **Solution:** Area = πr² = π(7)² = 49π cm² Circumference = 2πr = 2π(7) = 14π cm
  • **Example 2:** A circle has circumference 20π. Find its area. **Solution:** Step 1: C = 2πr → 20π = 2πr Step 2: r = 20π/(2π) = 10 Step 3: A = πr² = π(10)² = 100π
  • **Example 3:** Find the arc length of a 60° arc in a circle with radius 9. **Solution:** Arc length = (θ/360°) × 2πr = (60°/360°) × 2π(9) = (1/6) × 18π = 3π

Key Points

  • Circle = all points equidistant from center
  • Diameter = 2 × radius
  • Circumference = 2πr
  • Area = πr²
  • Tangent perpendicular to radius at contact point
  • Central angle = arc measure

**Inscribed Angle Theorem**: An inscribed angle is half the measure of its intercepted arc. If ∠ABC is inscribed and intercepts arc AC, then m∠ABC = (1/2) × m(arc AC) **Central Angle Theorem**: A central angle equals the measure of its intercepted arc. **Tangent-Chord Angle**: An angle formed by a tangent and chord equals half the intercepted arc. **Two-Chord Angle**: When two chords intersect inside a circle, the angle equals half the sum of the intercepted arcs. **Secant-Secant Angle**: When two secants intersect outside a circle, the angle equals half the difference of the intercepted arcs. **Power of a Point Theorems:** - **Two chords**: If chords AB and CD intersect at P, then AP × BP = CP × DP - **Secant-secant**: PA × PB = PC × PD (where P is external) - **Tangent-secant**: (PT)² = PA × PB (where PT is tangent)

Heading

Circle Theorems and Angle Relationships

Examples

  • **Example 1:** An inscribed angle intercepts an arc of 80°. Find the inscribed angle. **Solution:** Inscribed angle = (1/2) × arc = (1/2) × 80° = 40°
  • **Example 2:** Two chords intersect inside a circle. If the segments are 4, 6, and 8, find the fourth segment. **Solution:** Step 1: Use AP × BP = CP × DP Step 2: 4 × 6 = 8 × x Step 3: 24 = 8x → x = 3
  • **Example 3:** A tangent and secant from external point. If tangent = 12 and secant segments are 8 and x, find x. **Solution:** Step 1: (tangent)² = (near segment) × (whole secant) Step 2: 12² = 8 × (8 + x) Step 3: 144 = 64 + 8x Step 4: 80 = 8x → x = 10

Key Points

  • Inscribed angle = (1/2) × intercepted arc
  • Central angle = intercepted arc
  • Tangent perpendicular to radius
  • Two-chord angle = (1/2)(sum of arcs)
  • External angle = (1/2)(difference of arcs)
  • Power of point: product of segments

Mastering geometry requires knowing key formulas for perimeter, area, and volume. Here are the essential formulas organized by shape: **2D Shapes (Perimeter and Area):** **Rectangle**: P = 2(l + w), A = lw **Square**: P = 4s, A = s² **Triangle**: P = a + b + c, A = (1/2)bh **Circle**: C = 2πr, A = πr² **Trapezoid**: P = sum of all sides, A = (1/2)(b₁ + b₂)h **Parallelogram**: P = 2(a + b), A = bh **Rhombus**: P = 4s, A = (1/2)d₁d₂ **3D Shapes (Surface Area and Volume):** **Rectangular Prism**: SA = 2(lw + lh + wh), V = lwh **Cube**: SA = 6s², V = s³ **Cylinder**: SA = 2πr² + 2πrh, V = πr²h **Cone**: SA = πr² + πr√(r² + h²), V = (1/3)πr²h **Sphere**: SA = 4πr², V = (4/3)πr³ **Pyramid**: V = (1/3)Bh (B = base area)

Heading

Perimeter, Area, and Volume Formulas

Examples

  • **Example 1:** Find area and perimeter of rectangle with length 12 cm and width 8 cm. **Solution:** Perimeter = 2(l + w) = 2(12 + 8) = 2(20) = 40 cm Area = lw = 12 × 8 = 96 cm²
  • **Example 2:** A cylinder has radius 5 cm and height 10 cm. Find volume and surface area. **Solution:** Volume = πr²h = π(5)²(10) = 250π cm³ Surface Area = 2πr² + 2πrh = 2π(25) + 2π(5)(10) = 50π + 100π = 150π cm²
  • **Example 3:** A rectangular field's length is 5m less than 3 times its width. If perimeter is 54m, find dimensions. **Solution:** Step 1: Let w = width, then l = 3w - 5 Step 2: P = 2(l + w) → 54 = 2((3w - 5) + w) Step 3: 54 = 2(4w - 5) = 8w - 10 Step 4: 64 = 8w → w = 8m Step 5: l = 3(8) - 5 = 19m

Key Points

  • Perimeter = distance around shape
  • Area = space inside 2D shape
  • Volume = space inside 3D shape
  • Surface area = total outside area of 3D shape
  • Circle formulas use π ≈ 3.14159
  • Pyramid/cone volume = (1/3) × base area × height
Loading diagram…
Loading diagram…
Loading diagram…

Previous chapter

Word Problems — Number, Age, Work, Motion, Mixture, Investment

Next chapter

Perimeter, Area, Volume & Equation of a Line

Ready to practise for the USTET 2026?

Super Tutor's AI review plan adapts to your weak areas and builds a weekly practice schedule around your target USTET exam date.

Start free See the full USTET guide USTET preparation guide
← Back to Mathematics hub