Geometry: Lines, Angles, and Triangles (SAT)

Angles

Types of Angles

• Acute: $< 90°$
• Right: $= 90°$
• Obtuse: $> 90°$ and $< 180°$
• Straight: $= 180°$
• Reflex: $> 180°$

Angle Relationships

Complementary Angles: Sum to $90°$ $\text{If } \angle A + \angle B = 90°$

Supplementary Angles: Sum to $180°$ $\text{If } \angle A + \angle B = 180°$

Vertical Angles: Opposite angles when lines intersect (EQUAL)

Linear Pair: Adjacent angles on a straight line (sum to $180°$)

Parallel Lines and Transversals

When a line crosses two parallel lines:

Equal angles:

• Corresponding angles (same position)
• Alternate interior angles (inside, opposite sides)
• Alternate exterior angles (outside, opposite sides)

Supplementary angles:

• Consecutive interior angles (same side interior)

Triangles

Triangle Angle Sum

All triangles: angles sum to $180°$

$\angle A + \angle B + \angle C = 180°$

Types of Triangles

By angles:

• Acute: All angles $< 90°$
• Right: One angle $= 90°$
• Obtuse: One angle $> 90°$

By sides:

• Equilateral: All sides equal, all angles $60°$
• Isosceles: Two sides equal, two angles equal
• Scalene: No sides equal

Triangle Inequality

The sum of any two sides must be greater than the third side

If sides are $a$, $b$, $c$: $a + b > c$ $a + c > b$ $b + c > a$

Pythagorean Theorem

For right triangles only: $a^2 + b^2 = c^2$

Where $c$ is the hypotenuse (longest side, opposite right angle)

Common Pythagorean Triples

Memorize these for speed:

• 3-4-5 (and multiples: 6-8-10, 9-12-15)
• 5-12-13 (and multiples: 10-24-26)
• 8-15-17
• 7-24-25

Special Right Triangles

45-45-90 Triangle:

• Sides in ratio $x : x : x\sqrt{2}$
• If legs are 1, hypotenuse is $\sqrt{2}$
• If hypotenuse is $s$, legs are $\frac{s}{\sqrt{2}} = \frac{s\sqrt{2}}{2}$

30-60-90 Triangle:

• Sides in ratio $x : x\sqrt{3} : 2x$
• Opposite $30°$: shortest side ($x$)
• Opposite $60°$: middle side ($x\sqrt{3}$)
• Opposite $90°$: hypotenuse ($2x$)

Triangle Area

Standard formula: $A = \frac{1}{2}bh$

Where $b$ = base, $h$ = height (perpendicular to base)

For right triangles: $A = \frac{1}{2} \times \text{leg}_1 \times \text{leg}_2$

Exterior Angle Theorem

An exterior angle of a triangle equals the sum of the two remote interior angles

Similar Triangles

Definition: Same shape, different size (all corresponding angles equal)

Properties:

• Corresponding sides are proportional
• Ratio of areas = (ratio of sides)²

Example: If triangles are similar with ratio 2:3, then:

• Sides: $2:3$
• Perimeters: $2:3$
• Areas: $4:9$ (ratio squared!)

SAT Geometry Strategies

Draw and Label

Always draw the figure if not provided!

Look for Special Triangles

45-45-90 and 30-60-90 appear frequently

Use Pythagorean Theorem

Check if you can create right triangles

Mark Equal Angles and Sides

Visual marking helps spot relationships

Check for Similar Triangles

AA (two angles equal) proves similarity

Common SAT Traps

Trap 1: Assuming Figures Are Drawn to Scale

SAT says "figure not drawn to scale" - use given info only!

Trap 2: Forgetting Triangle Angle Sum

Always = $180°$, even if triangle looks weird

Trap 3: Mixing Up Special Triangles

30-60-90 vs 45-45-90 - check carefully!

Trap 4: Using Wrong Pythagorean Triple

Verify: $3^2 + 4^2 = 9 + 16 = 25 = 5^2$ ✓

Trap 5: Exterior Angle Confusion

Exterior angle = sum of TWO remote interior angles

SAT Tips

• Memorize special right triangles: 30-60-90 and 45-45-90
• Know Pythagorean triples: 3-4-5, 5-12-13, 8-15-17
• Triangle angles always sum to $180°$
• Vertical angles are equal
• Draw the figure if not given
• Label what you know on the diagram
• Look for parallel lines - lots of equal angles!
• Similar triangles: Corresponding sides are proportional