Grade 9 Geometry

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Supplementary Angles

Two angles that add up to 180° (think: Straight line). If one angle is 65°, the other is 180 − 65 = 115°.

Complementary Angles

Two angles that add up to 90° (think: Corner). If one angle is 65°, the other is 90 − 65 = 25°.

Triangle Angle Sum

All three interior angles of a triangle add to 180°. Formula: x + y + z = 180°

Exterior Angle Theorem

The exterior angle of a triangle equals the SUM of the two non-adjacent interior angles. Formula: a = x + y

Z-Pattern (Alternate Angles)

When two parallel lines are cut by a transversal, alternate angles (Z-shape) are EQUAL.

F-Pattern (Corresponding Angles)

When two parallel lines are cut by a transversal, corresponding angles (F-shape) are EQUAL.

C-Pattern (Co-Interior Angles)

When two parallel lines are cut by a transversal, co-interior angles (C-shape) ADD TO 180°. They are NOT equal.

Opposite (Vertically Opposite) Angles

When two lines cross, the angles directly across from each other are EQUAL. Adjacent angles are supplementary (180°).

Isosceles Triangle Theorem

An isosceles triangle has two equal sides AND two equal BASE angles. If base angle = 55°, apex = 180 − 55 − 55 = 70°.

Equilateral Triangle

All three sides equal, all three angles = 60°. Always. (3 × 60 = 180 ✓)

Pythagorean Theorem — Finding Hypotenuse

a² + b² = c². Example: legs = 6 and 8 → c = √(36+64) = √100 = 10. c is always the longest side opposite the right angle.

Pythagorean Theorem — Finding a Leg

a = √(c² − b²). Example: hyp = 13, leg = 5 → a = √(169−25) = √144 = 12. SUBTRACT, don't add.

Rectangle — Area & Perimeter

Area = l × w. Perimeter = 2l + 2w. Example: 9m × 5m → A = 45m², P = 28m.

Triangle — Area

A = (base × height) / 2. Example: base = 24cm, height = 10cm → A = 120cm².

Trapezoid — Area

A = (b₁ + b₂) / 2 × h. Example: parallel sides 10 and 24, height 12 → A = (34/2) × 12 = 204cm².

Circle — Area & Circumference

Area = πr². Circumference = 2πr (or πd). Example: r = 4.6 → A ≈ 66.5cm², C ≈ 28.9cm.

Semicircle — Area & Arc Length

Area = πr²/2. Arc length = πr (half the circumference). For perimeter of a semicircle shape: arc + diameter.

Composite Shape — Area

Break into known shapes, find each area separately, then ADD them together.

Composite Shape — Perimeter

Trace ONLY the outer boundary. Do NOT count internal seam lines where shapes join.

How Changing Dimensions Affects Perimeter

If you scale a dimension by factor k, perimeter (or circumference) scales by k. Double side → double perimeter.

How Changing Dimensions Affects Area

If you scale a dimension by factor k, area scales by k². Double both sides → area × 4, not × 2.

How Changing Dimensions Affects Volume

If you scale a dimension by factor k, volume scales by k³. Double all dimensions → volume × 8.

Volume of a Prism

V = Area of base × height. Find the base area first, then multiply by prism height. Example: base = 20cm², h = 6 → V = 120cm³.

Volume of a Pyramid

V = (Area of base × height) / 3. A pyramid is always 1/3 of the prism with the same base and height.

Volume of a Cylinder

V = πr²h. Example: r = 9.5cm, h = 6.4cm → V ≈ 1812.7cm³.

Volume of a Cone

V = πr²h / 3. A cone is always 1/3 of the cylinder with the same radius and height.

Volume of a Sphere

V = (4/3)πr³. Note: (4/3) not (1/3). Always use RADIUS, not diameter. Example: r = 5 → V ≈ 523.6.

Prism vs Pyramid Relationship

A pyramid with the same base and height as a prism has exactly 1/3 the volume. V_pyramid = V_prism ÷ 3.

Cylinder vs Cone Relationship

A cone with the same radius and height as a cylinder has exactly 1/3 the volume. V_cone = V_cylinder ÷ 3.