1. Identify and Classify Polygons

Definition: A polygon is a closed figure formed by three or more line segments that intersect only at their endpoints.

Classification by Number of Sides:

Classification by Properties:

Regular Polygon: All sides equal length AND all angles equal measure

Examples: Equilateral triangle, Square, Regular pentagon

Irregular Polygon: Sides or angles are NOT all equal

Examples: Scalene triangle, Rectangle (non-square)

Convex Polygon: All interior angles less than 180°

All vertices point outward

Concave Polygon: At least one interior angle greater than 180°

At least one vertex points inward (looks like it "caves in")

2. Classify Triangles

By Side Lengths:

Equilateral Triangle: All three sides equal

• All three angles equal (60° each)
• Regular polygon
• Notation: \( AB = BC = CA \)

Isosceles Triangle: Exactly two sides equal

• Two angles equal (base angles)
• Has a line of symmetry
• Notation: \( AB = AC \), \( \angle B = \angle C \)

Scalene Triangle: No sides equal

• No angles equal
• No line of symmetry
• All sides different lengths

By Angle Measures:

Acute Triangle: All three angles less than 90°

All angles acute: \( \angle A < 90° \), \( \angle B < 90° \), \( \angle C < 90° \)

Right Triangle: Exactly one angle equals 90°

• One right angle (90°)
• Two acute angles
• Sides: hypotenuse (longest), two legs
• Pythagorean theorem applies: \( a^2 + b^2 = c^2 \)

Obtuse Triangle: Exactly one angle greater than 90°

One obtuse angle, two acute angles

Combined Classifications:

Triangles can have both classifications: Right Isosceles Triangle, Obtuse Scalene Triangle, etc.

3. Identify Trapezoids

Definition: A trapezoid (trapezium) is a quadrilateral with exactly one pair of parallel sides.

Parts of a Trapezoid:

• Bases: The two parallel sides (usually horizontal)
• Legs: The two non-parallel sides
• Height: Perpendicular distance between the bases
• Midsegment: Line connecting midpoints of the legs

Types of Trapezoids:

Isosceles Trapezoid:

• Legs are equal in length
• Base angles are equal
• Diagonals are equal in length
• Has a line of symmetry

Right Trapezoid:

• Has two adjacent right angles (90°)
• One leg is perpendicular to both bases

Formulas:

Area: \( A = \frac{1}{2}(b_1 + b_2)h \)

where \( b_1 \) and \( b_2 \) are the bases, \( h \) is the height

Midsegment: \( m = \frac{b_1 + b_2}{2} \)

The midsegment is parallel to the bases and equals their average

Perimeter: \( P = b_1 + b_2 + l_1 + l_2 \)

4. Classify Quadrilaterals I & II

Definition: A quadrilateral is a polygon with four sides and four angles.

Types of Quadrilaterals:

Quadrilateral Hierarchy:

Square → is a Rectangle AND a Rhombus

Rectangle and Rhombus → are both Parallelograms

All Parallelograms → are Quadrilaterals

5. Area and Perimeter Formulas for Quadrilaterals

Key: \( s \) = side, \( l \) = length, \( w \) = width, \( b \) = base, \( h \) = height, \( d_1, d_2 \) = diagonals

6. Triangle Angle-Sum Theorem

Theorem: The sum of the measures of the interior angles of a triangle is always 180°.

\( \angle A + \angle B + \angle C = 180° \)

Examples:

Example 1: Two angles of a triangle are 50° and 70°. Find the third angle.

\( 50° + 70° + x = 180° \)

\( 120° + x = 180° \)

\( x = 60° \)

Example 2: In a triangle, the angles are in the ratio 2:3:4. Find all three angles.

Let angles be \( 2x, 3x, 4x \)

\( 2x + 3x + 4x = 180° \)

\( 9x = 180° \) → \( x = 20° \)

Angles: 40°, 60°, 80°

Example 3: Find the missing angle if two angles are \( (2x + 10)° \) and \( (3x - 5)° \), and the third is \( 75° \).

\( (2x + 10) + (3x - 5) + 75 = 180 \)

\( 5x + 80 = 180 \) → \( 5x = 100 \) → \( x = 20 \)

Angles: 50°, 55°, 75°

7. Exterior Angle Theorem

Theorem: An exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles.

\( \text{Exterior Angle} = \angle A + \angle B \)

Key Points:

• An exterior angle is formed when one side of a triangle is extended
• Each vertex of a triangle has an exterior angle
• Exterior angle + adjacent interior angle = 180° (linear pair)
• Sum of all exterior angles of a triangle = 360°

Examples:

Example 1: Two interior angles of a triangle are 40° and 60°. Find the exterior angle at the third vertex.

Method 1: Exterior angle = \( 40° + 60° = 100° \)

Method 2: Third interior angle = \( 180° - 40° - 60° = 80° \)

Exterior angle = \( 180° - 80° = 100° \)

Example 2: An exterior angle is 120°. If one remote interior angle is 50°, find the other remote interior angle.

\( 120° = 50° + x \)

\( x = 70° \)

Example 3: An exterior angle is \( (5x + 10)° \). The two remote interior angles are \( (2x + 5)° \) and \( (3x)° \). Find \( x \).

\( 5x + 10 = (2x + 5) + 3x \)

\( 5x + 10 = 5x + 5 \)

This would lead to \( 10 = 5 \) (contradiction - check problem setup)

8. Find Missing Angles in Quadrilaterals

Key Fact: The sum of interior angles of any quadrilateral is 360°.

\( \angle A + \angle B + \angle C + \angle D = 360° \)

Special Properties:

• Rectangle/Square: All angles = 90°
• Parallelogram/Rhombus: Opposite angles are equal
• Isosceles Trapezoid: Base angles are equal
• Kite: One pair of opposite angles are equal

Examples:

Example 1: Three angles of a quadrilateral are 80°, 110°, and 75°. Find the fourth angle.

\( 80° + 110° + 75° + x = 360° \)

\( 265° + x = 360° \)

\( x = 95° \)

Example 2: In a parallelogram, one angle is 65°. Find all other angles.

Opposite angle: 65° (opposite angles equal)

Adjacent angles: \( 180° - 65° = 115° \) each (consecutive angles supplementary)

All angles: 65°, 115°, 65°, 115°

Example 3: In a trapezoid, angles are in ratio 2:3:4:6. Find all angles.

Let angles be \( 2x, 3x, 4x, 6x \)

\( 2x + 3x + 4x + 6x = 360° \)

\( 15x = 360° \) → \( x = 24° \)

Angles: 48°, 72°, 96°, 144°

9. Interior Angles of Polygons

Key Formulas:

Sum of Interior Angles: \( S = (n - 2) \times 180° \)

where \( n \) = number of sides

Each Interior Angle (Regular Polygon): \( \text{Each angle} = \frac{(n-2) \times 180°}{n} \)

Sum of Exterior Angles: Always 360° (for any polygon)

Each Exterior Angle (Regular Polygon): \( \text{Each angle} = \frac{360°}{n} \)

Quick Reference Table:

Examples:

Example 1: Find the sum of interior angles of a 10-sided polygon.

\( S = (10 - 2) \times 180° = 8 \times 180° = 1440° \)

Example 2: Each interior angle of a regular polygon is 140°. How many sides does it have?

\( \frac{(n-2) \times 180}{n} = 140 \)

\( (n-2) \times 180 = 140n \)

\( 180n - 360 = 140n \)

\( 40n = 360 \) → \( n = 9 \) (nonagon)

10. Parts of a Circle

Basic Components:

Center: The fixed point in the middle of the circle (usually denoted as O)

Radius (r): Distance from center to any point on the circle

• All radii of a circle are equal
• Formula: \( r = \frac{d}{2} \)

Diameter (d): Line segment passing through center with endpoints on circle

• Longest chord in a circle
• Formula: \( d = 2r \)

Chord: Line segment with both endpoints on the circle

• Diameter is the longest chord
• Does not have to pass through center

Circumference (C): Distance around the circle (perimeter)

\( C = 2\pi r = \pi d \)

Arc: Part of the circumference between two points

• Minor arc: Shorter arc (less than 180°)
• Major arc: Longer arc (greater than 180°)
• Semicircle: Arc equal to 180° (half circle)

Secant: Line that intersects the circle at two points

Tangent: Line that touches the circle at exactly one point

• Perpendicular to radius at point of tangency
• Point of tangency: where tangent touches circle

Central Angle: Angle with vertex at the center

• Measure equals the arc it intercepts

Inscribed Angle: Angle with vertex on the circle

• Measure is half the intercepted arc

Area and Circumference Formulas:

Area: \( A = \pi r^2 \)

Circumference: \( C = 2\pi r = \pi d \)

Examples:

Example 1: A circle has radius 7 cm. Find circumference and area.

\( C = 2\pi(7) = 14\pi \approx 43.98 \) cm

\( A = \pi(7)^2 = 49\pi \approx 153.94 \) cm²

Example 2: A circle has diameter 20 m. Find the radius and area.

\( r = \frac{20}{2} = 10 \) m

\( A = \pi(10)^2 = 100\pi \approx 314.16 \) m²

Quick Reference: Essential Formulas

Triangle Formulas:

• Sum of angles: \( 180° \)
• Exterior angle = Sum of two remote interior angles
• Area: \( A = \frac{1}{2}bh \)

Quadrilateral Formulas:

• Sum of angles: \( 360° \)
• Square: \( A = s^2 \), \( P = 4s \)
• Rectangle: \( A = lw \), \( P = 2(l+w) \)
• Trapezoid: \( A = \frac{1}{2}(b_1 + b_2)h \)

Polygon Formulas:

• Sum of interior angles: \( (n-2) \times 180° \)
• Each interior angle (regular): \( \frac{(n-2) \times 180°}{n} \)
• Sum of exterior angles: \( 360° \)
• Each exterior angle (regular): \( \frac{360°}{n} \)

Circle Formulas:

• Circumference: \( C = 2\pi r = \pi d \)
• Area: \( A = \pi r^2 \)
• Diameter: \( d = 2r \)

💡 Key Tips for Two-Dimensional Figures

• ✓ Triangle angles always sum to 180° (no exceptions!)
• ✓ Quadrilateral angles always sum to 360°
• ✓ Exterior angle of triangle = sum of two remote interior angles
• ✓ All squares are rectangles, but not all rectangles are squares
• ✓ All squares are rhombuses, but not all rhombuses are squares
• ✓ Trapezoid has exactly one pair of parallel sides
• ✓ Regular polygons have all sides and angles equal
• ✓ Sum of exterior angles is always 360° for any polygon
• ✓ Use formula (n-2)×180° for sum of interior angles
• ✓ Diameter = 2 × radius in circles
• ✓ Remember π ≈ 3.14 for circle calculations