ACT Geometry Crunch 🎯 | Algebra II ACT Prep

Finding Angles in Transversal Problems (Level 2)

Triangles

📐 Key Rules — Parallel Lines Cut by a Transversal

Corresponding angles are EQUAL (a = e, b = f, c = g, d = h)
Alternate interior angles are EQUAL (c = f, d = e)
Alternate exterior angles are EQUAL (a = h, b = g)
Co-interior (same-side) angles are SUPPLEMENTARY (c + e = 180°)

📝 Worked Example

Two parallel lines are cut by a transversal. One angle is (3x + 20)° and its alternate interior angle is (5x − 40)°. Find x and both angles.

1Alternate interior angles are equal → set them equal: 3x + 20 = 5x − 40

2Subtract 3x: 20 = 2x − 40

3Add 40: 60 = 2x → x = 30

4Check: 3(30)+20 = 110° and 5(30)−40 = 110° ✓

✓ x = 30, both alternate interior angles = 110°

⚡ACT Tip: The lines will almost always be parallel. If the angles are on the same side of the transversal between the parallel lines, they add to 180°. Otherwise they're equal. When in doubt: equal OR supplementary.

Solve for Interior Angles – Triangle (Level 2)

Triangles

📐 Key Rule

The three interior angles of ANY triangle always sum to 180°

∠A + ∠B + ∠C = 180°

📝 Worked Example

The three angles of a triangle are (x + 10)°, (2x + 5)°, and (x + 25)°. Find x and each angle.

1Set sum equal to 180: (x+10) + (2x+5) + (x+25) = 180

2Combine like terms: 4x + 40 = 180

3Subtract 40, divide by 4: 4x = 140 → x = 35

4Angles: 45°, 75°, 60° Check: 45+75+60 = 180 ✓

✓ x = 35 → angles are 45°, 75°, 60°

⚡ACT Tip: Always combine ALL constant terms before solving — forgetting to collect them is the most common mistake. After solving, add all three angles to confirm they equal 180°.

Exterior Angles (Triangle)

Triangles

📐 Key Rule

An exterior angle of a triangle equals the SUM of the two non-adjacent interior angles (called the "remote interior angles").

Exterior angle = Remote Interior₁ + Remote Interior₂

📝 Worked Example

The exterior angle of a triangle is (4x + 10)°. The two remote interior angles are (x + 30)° and (2x − 5)°. Find x.

1Set up: exterior = sum of remote interiors

24x + 10 = (x + 30) + (2x − 5) → 4x + 10 = 3x + 25

3Subtract 3x: x + 10 = 25 → x = 15

4Exterior: 4(15)+10 = 70°. Interiors: 45° + 25° = 70° ✓

✓ x = 15, exterior angle = 70°

⚡ACT Tip: The exterior angle is always larger than either remote interior angle — if it comes out smaller, your equation is set up wrong. Also remember: the exterior angle + its adjacent interior angle = 180° (they form a straight line).

The Triangle Inequality

Triangles

📐 Key Rule

The sum of any TWO sides must be GREATER THAN the third side.

If you know sides a and b, the third side x must satisfy:

|a − b| < x < a + b

The third side is strictly between the difference and the sum.

📝 Worked Example

Two sides of a triangle measure 7 and 11. What are the possible lengths of the third side?

1Lower bound: |7 − 11| = |−4| = 4 → third side must be GREATER THAN 4

2Upper bound: 7 + 11 = 18 → third side must be LESS THAN 18

3Write the range: 4 < x < 18

✓ Third side must satisfy: 4 < x < 18

⚡ACT Tip: The bounds are strict — the side cannot equal 4 or 18 exactly (that would make a flat, degenerate triangle). The ACT often asks which answer choice CANNOT be the third side — quickly check if any option is ≤ 4 or ≥ 18.

Find Slope / Y-Intercept from Equation (Standard Form)

Lines

📐 Key Rules

Standard Form: Ax + By = C

Solve for y to get slope-intercept (y = mx + b):

Slope: m = −A/B
Y-intercept: b = C/B

Or just rearrange: subtract Ax, divide by B.

📝 Worked Example

Find the slope and y-intercept of 3x + 4y = 12.

1Subtract 3x from both sides: 4y = −3x + 12

2Divide everything by 4: y = −¾x + 3

3Identify: slope m = −3/4 and y-intercept b = 3

✓ Slope = −3/4 | Y-intercept = (0, 3)

⚡ACT Speed Tip: Use the shortcut — slope = −A/B. For 3x + 4y = 12: slope = −3/4. Don't forget the negative sign — it's extremely easy to miss! Y-intercept shortcut = C/B = 12/4 = 3.

The Midpoint Formula

Lines

📐 Key Formula

Midpoint M = ( (x₁ + x₂) / 2 , (y₁ + y₂) / 2 )

→ Average the x-coordinates, average the y-coordinates.

Finding missing endpoint: multiply midpoint × 2, subtract known point.

📝 Worked Example

Find the midpoint of the segment with endpoints (2, −5) and (8, 3).

1Average the x-values: (2 + 8) / 2 = 10 / 2 = 5

2Average the y-values: (−5 + 3) / 2 = −2 / 2 = −1

✓ Midpoint = (5, −1)

⚡ACT Tip: The ACT often gives you the midpoint and one endpoint and asks for the OTHER endpoint. Work backwards: if midpoint = (5, −1) and one point = (2, −5), multiply midpoint coords by 2 and subtract: other point = (2·5 − 2, 2·(−1) − (−5)) = (8, 3).

Parallel and Perpendicular Equations

Lines

📐 Key Rules

Parallel lines → SAME slope, different y-intercept (m₁ = m₂)
Perpendicular lines → NEGATIVE RECIPROCAL slopes (m₁ × m₂ = −1)

Flip the fraction AND change the sign to get ⊥ slope.

Point-Slope Form (to write the equation): y − y₁ = m(x − x₁)

📝 Worked Example

Write equations for lines through (2, 4) that are: (a) parallel to y = 3x + 1 and (b) perpendicular to y = 3x + 1

1(a) Parallel → same slope m = 3. Point-slope: y − 4 = 3(x − 2)

2Simplify: y = 3x − 6 + 4 → y = 3x − 2

3(b) Perpendicular → flip and negate: slope = −1/3. y − 4 = −⅓(x − 2)

4Simplify: y = −⅓x + ⅔ + 4 → y = −⅓x + 14/3

✓ Parallel: y = 3x − 2 | Perpendicular: y = −⅓x + 14/3

⚡ACT Tip: To get a perpendicular slope: flip the fraction and change the sign. Slope 2/3 → perpendicular is −3/2. Slope 5 (= 5/1) → perpendicular is −1/5. Quick sanity check: multiply the two slopes — you should get −1.

Find Circle Center and Radius from Equation (Conic Form)

Circles

📐 Key Formula

Standard (conic) form of a circle:

   (x − h)² + (y − k)² = r²

   Center = (h, k)    Radius = r = √(r²)

⚠️ SIGN TRAP: (y + 2)² means k = −2, NOT +2
   Because (y + 2) = (y − (−2)) → k = −2

📝 Worked Example

Find the center and radius: (x − 3)² + (y + 2)² = 25

1Compare to (x − h)² + (y − k)² = r²

2(x − 3)² → h = 3. (y + 2)² = (y − (−2))² → k = −2

3r² = 25 → r = √25 = 5

✓ Center: (3, −2) | Radius: 5

⚡ACT Tip: The most common mistake is the sign of the center. If you see (x + 5)², the center's x-coordinate is negative 5. The formula has "x − h", so (x + 5) = (x − (−5)) means h = −5. Always ask yourself: "what would make this expression equal zero?"

Circumference (in terms of π)

Circles

📐 Key Formulas

C = 2πr (using radius r)
C = πd (using diameter d, where d = 2r)

Leave answer with π — do NOT substitute 3.14!

📝 Worked Example

Find the circumference of a circle with radius 7. Leave in terms of π.

1Use C = 2πr with r = 7

2C = 2π(7) = 14π

✓ C = 14π

⚡ACT Tip: Always check — did they give you the radius or the diameter? They'll often give the diameter on purpose to trip you up. If you see "diameter = 10," the radius is 5. C = πd = π(10) = 10π.

Area of a Circle (in terms of π)

Circles

📐 Key Formula

A = πr²

If given diameter: r = d ÷ 2, then A = π(d/2)²

Leave answer with π!

📝 Worked Example

A circle has a diameter of 10. Find its area in terms of π.

1Diameter = 10, so radius r = 10 ÷ 2 = 5

2A = πr² = π(5)² = π(25) = 25π

✓ A = 25π

⚡ACT Tip: Square the radius FIRST, then multiply by π. Don't skip steps. Common error: (2r)² = 4r², not 2r². Easy memory: Area uses r² (two letters, exponent of 2). Circumference uses r (one letter, no exponent).

Circumference and Area of a Circle (Combined)

Circles

📐 Key Strategy

C = 2πr and A = πr²

When given C or A and asked for the other:
Step 1: Find r. Step 2: Use the other formula.

📝 Worked Example

A circle has a circumference of 16π. Find its area in terms of π.

1Use C = 2πr: 16π = 2πr

2Divide both sides by 2π: r = 8

3Now find area: A = πr² = π(8)² = 64π

✓ A = 64π

⚡ACT Tip: Always find r as the middle step — don't try to jump directly from circumference to area. Once you have r, both formulas are quick. The π cancels nicely when solving for r from circumference.

Central / Inscribed Angles – Level 1

Circles

📐 Key Rules

Central angle = intercepted arc (they're equal)
Inscribed angle = ½ × intercepted arc

→ Same arc: Inscribed angle = ½ × Central angle

📝 Worked Example

A central angle measures 80°. An inscribed angle intercepts the same arc. Find the inscribed angle.

1Central angle = arc = 80°

2Inscribed angle = ½ × arc = ½ × 80° = 40°

✓ Inscribed angle = 40°

⚡ACT Tip: An inscribed angle in a semicircle (arc = 180°) always equals 90°. This is a super common ACT setup — if a triangle is inscribed in a circle with one side as the diameter, the angle opposite that side is always exactly 90°.

Central / Inscribed Angles (Algebraic)

Circles

📐 Key Relationship

Central angle = 2 × Inscribed angle (same arc)

→ Set up that equation, then solve for the variable.
→ Always verify: central should be exactly double the inscribed.

📝 Worked Example

An inscribed angle measures (2x + 10)°. The central angle intercepting the same arc measures (6x − 20)°. Find x and both angles.

1Central = 2 × Inscribed → 6x − 20 = 2(2x + 10)

2Distribute: 6x − 20 = 4x + 20

3Subtract 4x: 2x = 40 → x = 20

4Inscribed: 2(20)+10 = 50°. Central: 6(20)−20 = 100° = 2×50 ✓

✓ x = 20 | Inscribed = 50° | Central = 100°

⚡ACT Tip: After solving, always do a quick check — is the central angle exactly double the inscribed angle? If not, you set up the equation backwards. The relationship is always Central = 2 × Inscribed.

Angles Formed by Chords, Tangents, Secants (Level 1)

Circles

📐 Key Rules — WHERE is the vertex?

Vertex INSIDE the circle (two chords crossing):
   angle = ½ × (arc₁ + arc₂)   ← ADD the two arcs

Vertex OUTSIDE the circle (two secants/tangents):
   angle = ½ × (far arc − near arc)   ← SUBTRACT the arcs

Vertex ON the circle (tangent-chord):
   angle = ½ × intercepted arc   (same as inscribed angle)

📝 Worked Examples

Example A — Inside: Two chords intersect inside a circle. Intercepted arcs are 80° and 60°. Find the angle.

1Vertex is inside → add arcs, divide by 2: angle = ½(80° + 60°) = ½(140°)

✓ Angle = 70°

Example B — Outside: Two secants from an external point. Far arc = 120°, near arc = 40°. Find the angle.

1Vertex is outside → subtract arcs, divide by 2: angle = ½(120° − 40°) = ½(80°)

✓ Angle = 40°

⚡ACT Memory Trick: INSIDE → ADD arcs ÷ 2. OUTSIDE → SUBTRACT arcs ÷ 2. ON → half the arc. The key is always WHERE the vertex is. Look at that first before doing any math.

Simple Proportions

Proportions

📐 Key Method

a/b = c/d → Cross multiply: a × d = b × c

Then solve the resulting equation for the unknown.

📝 Worked Example

Solve: x/5 = 12/20

1Cross multiply: 20 · x = 5 · 12 → 20x = 60

2Divide by 20: x = 60 ÷ 20 = 3

✓ x = 3

⚡ACT Speed Tip: If you can simplify the fraction first, do it — the numbers will be smaller and easier. For x/5 = 12/20: simplify 12/20 = 3/5. Now x/5 = 3/5, so x = 3 by inspection — no cross-multiplying needed!

Solving Linear Proportions

Proportions

📐 Key Method

Same as simple proportions — cross multiply.
Difference: one or both sides have expressions like (x + 2) or (2x − 1).

After cross-multiplying: distribute, collect like terms, solve for x.

📝 Worked Example

Solve: (x + 2)/3 = (2x − 1)/5

1Cross multiply: 5(x + 2) = 3(2x − 1)

2Distribute both sides: 5x + 10 = 6x − 3

3Subtract 5x from both sides: 10 = x − 3 → x = 13

4Check: (13+2)/3 = 15/3 = 5 and (26−1)/5 = 25/5 = 5 ✓

✓ x = 13

⚡ACT Tip: After cross-multiplying, distribute the number to BOTH terms in the parentheses — the most common error is only distributing to the first term. Always plug your answer back into both sides to confirm they match.