The Study of Shape, Space, and Structure
These are the basic building blocks that cannot be defined using simpler terms:
| Term | Definition | Notation |
|---|---|---|
| Line Segment | Part of a line with two endpoints | AB or BA |
| Ray | Part of a line with one endpoint, extends infinitely | AB→ |
| Angle | Two rays sharing a common endpoint (vertex) | ∠ABC or ∠B |
| Collinear Points | Points that lie on the same line | - |
| Coplanar Points | Points that lie on the same plane | - |
If B is between A and C, then AB + BC = AC
If D is in the interior of ∠ABC, then ∠ABD + ∠DBC = ∠ABC
1. If AB = 5 and BC = 8, and B is between A and C, find AC.
AC =
2. If ∠ABD = 35° and ∠DBC = 47°, find ∠ABC.
∠ABC = °
3. Point M is the midpoint of AB. If AM = 7, find AB.
AB =
Inductive Reasoning: Making a conjecture based on patterns observed
Example: 2, 4, 6, 8, ... → Next is probably 10
Deductive Reasoning: Using facts, definitions, and logic to prove something
Example: All mammals breathe air. Dogs are mammals. ∴ Dogs breathe air.
If-Then form: If p, then q (written p → q)
| Statement | Form | Example |
|---|---|---|
| Conditional | p → q | If it rains, then the ground is wet |
| Converse | q → p | If the ground is wet, then it rains |
| Inverse | ~p → ~q | If it doesn't rain, then the ground is not wet |
| Contrapositive | ~q → ~p | If the ground is not wet, then it doesn't rain |
Note: A conditional and its contrapositive have the same truth value!
| Statements | Reasons |
|---|---|
| 1. Given information | Given |
| 2. Step toward conclusion | Definition/Postulate/Theorem |
| 3. ... | ... |
| n. What you're proving | Final reason |
1. Write the converse of: "If a figure is a square, then it has four sides."
2. Write the contrapositive of: "If x = 5, then x² = 25."
| Angle Pair | Location | If Lines Parallel |
|---|---|---|
| Corresponding Angles | Same position at each intersection | Congruent |
| Alternate Interior Angles | Inside, opposite sides of transversal | Congruent |
| Alternate Exterior Angles | Outside, opposite sides of transversal | Congruent |
| Same-Side Interior Angles | Inside, same side of transversal | Supplementary (180°) |
If two parallel lines are cut by a transversal, then:
1. Lines m ∥ n are cut by transversal t. If ∠1 = 72°, find the following:
Corresponding angle to ∠1: °
Alternate interior angle to ∠1: °
Same-side interior angle to ∠1: °
2. If two lines are perpendicular, all four angles formed are °
| Method | What Must Match |
|---|---|
| SSS (Side-Side-Side) | All three sides |
| SAS (Side-Angle-Side) | Two sides and the included angle |
| ASA (Angle-Side-Angle) | Two angles and the included side |
| AAS (Angle-Angle-Side) | Two angles and a non-included side |
| HL (Hypotenuse-Leg) | Right triangles: hypotenuse and one leg |
WARNING: SSA and AAA do NOT prove congruence!
Corresponding Parts of Congruent Triangles are Congruent
Once you prove triangles congruent, ALL corresponding parts are congruent!
1. Which congruence postulate/theorem would you use?
a) Two sides and the angle between them are congruent:
b) All three sides are congruent:
c) Two angles and the side between them:
2. △ABC ≅ △DEF. If AB = 5, BC = 7, and AC = 9, find DF.
DF =
The sum of the interior angles of a triangle is 180°
∠A + ∠B + ∠C = 180°
An exterior angle equals the sum of the two non-adjacent interior angles
By Sides:
By Angles:
The sum of any two sides must be greater than the third side.
a + b > c, a + c > b, b + c > a
1. Two angles of a triangle measure 45° and 72°. Find the third angle.
Third angle = °
2. Can a triangle have sides of length 3, 4, and 8? (Yes/No)
Why?
3. An exterior angle of a triangle is 120°. If one non-adjacent interior angle is 45°, find the other.
Other interior angle = °
The sum of interior angles of any quadrilateral is 360°
| Shape | Properties |
|---|---|
| Parallelogram | Opposite sides parallel and congruent; opposite angles congruent; diagonals bisect each other |
| Rectangle | Parallelogram + all angles 90°; diagonals congruent |
| Rhombus | Parallelogram + all sides congruent; diagonals perpendicular |
| Square | Rectangle + Rhombus (all sides congruent, all angles 90°) |
| Trapezoid | Exactly one pair of parallel sides (bases) |
| Kite | Two pairs of consecutive congruent sides |
1. Three angles of a quadrilateral are 85°, 100°, and 75°. Find the fourth angle.
Fourth angle = °
2. True or False:
a) Every square is a rectangle:
b) Every rectangle is a square:
c) Every rhombus is a parallelogram:
Similar figures have the same shape but not necessarily the same size.
We write: △ABC ~ △DEF
| Method | Requirements |
|---|---|
| AA (Angle-Angle) | Two pairs of congruent angles |
| SSS (Side-Side-Side) | All three pairs of sides proportional |
| SAS (Side-Angle-Side) | Two pairs of proportional sides and included angle congruent |
If △ABC ~ △DEF with scale factor k, then:
AB/DE = BC/EF = AC/DF = k
1. △ABC ~ △DEF. AB = 6, DE = 9, BC = 8. Find EF.
EF =
2. What is the scale factor from △ABC to △DEF in problem 1?
Scale factor =
In a right triangle: a² + b² = c²
where c is the hypotenuse (longest side, opposite the right angle)
| Triangle | Angle Measures | Side Ratios |
|---|---|---|
| 45-45-90 | 45°, 45°, 90° | 1 : 1 : √2 |
| 30-60-90 | 30°, 60°, 90° | 1 : √3 : 2 |
sin θ = Opposite / Hypotenuse
cos θ = Adjacent / Hypotenuse
tan θ = Opposite / Adjacent
In a right triangle, one leg = 5, the other leg = 12. Find the hypotenuse.
1. Find the missing side:
a) Legs: 3 and 4. Hypotenuse =
b) Leg: 8, Hypotenuse: 17. Other leg =
2. In a 45-45-90 triangle, if the legs are 7, the hypotenuse is
3. In a right triangle, the side opposite a 30° angle is 5. The hypotenuse is
Circumference: C = 2πr = πd
Area: A = πr²
Arc Length: L = (θ/360°) × 2πr
Sector Area: A = (θ/360°) × πr²
1. A circle has radius 7. Find:
Diameter =
Circumference = π
Area = π
2. A central angle is 80°. The inscribed angle intercepting the same arc is °
| Shape | Perimeter | Area |
|---|---|---|
| Square | P = 4s | A = s² |
| Rectangle | P = 2l + 2w | A = lw |
| Triangle | P = a + b + c | A = ½bh |
| Parallelogram | P = 2a + 2b | A = bh |
| Trapezoid | P = sum of sides | A = ½(b₁ + b₂)h |
| Regular Polygon | P = ns | A = ½ap |
| Circle | C = 2πr | A = πr² |
1. Rectangle: length = 12, width = 5
Perimeter = Area =
2. Triangle: base = 10, height = 6
Area =
3. Trapezoid: bases = 8 and 12, height = 5
Area =
| Solid | Surface Area | Volume |
|---|---|---|
| Cube | SA = 6s² | V = s³ |
| Rectangular Prism | SA = 2(lw + lh + wh) | V = lwh |
| Cylinder | SA = 2πr² + 2πrh | V = πr²h |
| Cone | SA = πr² + πrl | V = ⅓πr²h |
| Sphere | SA = 4πr² | V = ⁴⁄₃πr³ |
| Pyramid | SA = base + ½Pl | V = ⅓Bh |
1. Cube with side 4:
Surface Area = Volume =
2. Cylinder: r = 3, h = 10
Volume = π
3. Sphere: r = 6
Volume = π
| Transformation | Description | Preserves |
|---|---|---|
| Translation | Slide (every point moves same distance/direction) | Size, Shape |
| Reflection | Flip (mirror image over a line) | Size, Shape |
| Rotation | Turn around a point | Size, Shape |
| Dilation | Enlarge or shrink from a center point | Shape only |
Translation: (x, y) → (x + a, y + b)
Reflection over x-axis: (x, y) → (x, -y)
Reflection over y-axis: (x, y) → (-x, y)
Rotation 90° counterclockwise: (x, y) → (-y, x)
Rotation 180°: (x, y) → (-x, -y)
Dilation with scale factor k: (x, y) → (kx, ky)
1. Point A(3, 5) is translated by (x + 2, y - 4). New coordinates:
A' = (, )
2. Point B(4, -2) is reflected over the x-axis. New coordinates:
B' = (, )
3. Point C(2, 6) is dilated with scale factor 3. New coordinates:
C' = (, )
Unit 1: 1) 13, 2) 82°, 3) 14
Unit 3: 1) 72°, 72°, 108°; 2) 90°
Unit 4: 1a) SAS, b) SSS, c) ASA; 2) 9
Unit 5: 1) 63°, 2) No (3+4=7<8), 3) 75°
Unit 6: 1) 100°; 2a) True, b) False, c) True
Unit 7: 1) 12; 2) 3/2 or 1.5
Unit 8: 1a) 5, b) 15; 2) 7√2; 3) 10
Unit 9: 1) d=14, C=14π, A=49π; 2) 40°
Unit 10: 1) P=34, A=60; 2) 30; 3) 50
Unit 11: 1) SA=96, V=64; 2) 90π; 3) 288π
Unit 12: 1) (5,1); 2) (4,2); 3) (6,18)