Angles & Geometry

4.J.1Lines, rays, segments01/04

Lines, rays, segments; parallel and perpendicular.

Step01of 04

Three basic figures, distinguished by how far they extend.

Point	→a single location, no length
Line	→extends infinitely in both directions
Ray	→starts at one point, extends infinitely in one direction
Segment	→finite — has TWO endpoints

4.J.1 — Lines, rays, segments; parallel and perpendicular.

Step 01 of 04

Three basic figures, distinguished by how far they extend.

Point	→a single location, no length
Line	→extends infinitely in both directions
Ray	→starts at one point, extends infinitely in one direction
Segment	→finite — has TWO endpoints

Step 02 of 04

Pairs of lines.

Parallel lines	→never cross — always the same distance apart (symbol ∥)
Perpendicular lines	→cross at a RIGHT angle (90°) — symbol ⊥
Intersecting lines	→cross at any angle — could be perpendicular or just oblique

Step 03 of 04

Real-world examples.

Railroad tracks	→parallel
+ sign	→perpendicular (vertical and horizontal lines)
Floor and wall	→perpendicular (where they meet)
Highway lanes	→parallel
X mark	→intersecting (not perpendicular)

Step 04 of 04

Notation.

$\overline{A B}$	→segment AB
$A B$	→ray AB (starts at A)
$A B$	→line AB
$A B ∥ C D$	→AB parallel to CD
$A B ⊥ C D$	→AB perpendicular to CD

Key insight

Line goes forever both ways. Ray goes one way. Segment has two endpoints. Parallel lines never meet; perpendicular lines meet at 90°.

4.J.2 — Classify triangles and quadrilaterals.

Step 01 of 04

Triangles by angles.

Acute	→all three angles less than 90°
Right	→one angle exactly 90°
Obtuse	→one angle GREATER than 90°

Triangles by sides.

Equilateral	→all 3 sides equal
Isosceles	→2 sides equal
Scalene	→all 3 sides different

Step 02 of 04

A triangle can be classified BOTH ways. "Right isosceles triangle" = has a right angle AND two equal sides.

Step 03 of 04

Quadrilaterals. Four-sided shapes. Each special type adds an extra property.

Quadrilateral	→any 4-sided shape
Trapezoid	→exactly one pair of parallel sides
Parallelogram	→BOTH pairs of opposite sides parallel
Rectangle	→parallelogram with 4 right angles
Rhombus	→parallelogram with 4 equal sides
Square	→rectangle AND rhombus (4 right angles + 4 equal sides)

Step 04 of 04

Hierarchy. Every square is a rectangle, AND a rhombus, AND a parallelogram. Every rectangle is a parallelogram. Working up the chain, each adds more properties.

Key insight

Triangles classified by angles (acute/right/obtuse) AND by sides (equilateral/isosceles/scalene). Quadrilaterals form a hierarchy: square ⊂ rectangle ⊂ parallelogram ⊂ quadrilateral.

4.J.3 — Lines of symmetry.

Step 01 of 04

A line of symmetry is a line that divides a shape into TWO MIRROR IMAGES. If you fold along the line, the two halves match exactly.

Step 02 of 04

Common shapes and their lines of symmetry.

Equilateral triangle	→3 (one through each vertex to the opposite side)
Isosceles triangle	→1
Scalene triangle	→0
Square	→4 (2 diagonals + 2 perpendicular bisectors)
Rectangle (non-square)	→2 (through midpoints of opposite sides)
Regular hexagon	→6
Circle	→INFINITE (every diameter)

Step 03 of 04

Letters of the alphabet.

A, M, T, U, V, W, Y	→one vertical line of symmetry
B, C, D, E, K	→one horizontal line of symmetry
H, I, O, X	→two or more lines of symmetry
F, G, J, L, N, P, Q, R, S, Z	→no lines of symmetry

Step 04 of 04

How to test. Imagine folding the shape along a candidate line. If both halves overlap exactly, it's a line of symmetry.

Most regular polygons have lines of symmetry equal to their number of sides. Most irregular shapes have 0 or 1.

Key insight

Line of symmetry = imaginary fold line that makes halves match. Regular polygons (equal sides AND angles) have the same number of symmetry lines as sides. Circles have infinite.

4.J.4 — Measure angles and angle sums.

Step 01 of 04

Angles measure rotation, in degrees. A full circle = 360°.

Right angle	→90°
Straight angle	→180°
Acute angle	→less than 90°
Obtuse angle	→between 90° and 180°
Reflex angle	→greater than 180°

Step 02 of 04

Using a protractor.

1.	→place center on the angle's vertex
2.	→align one ray with the 0° line
3.	→read where the other ray crosses — use the scale starting at 0

Protractors usually have TWO scales. Pick the one that starts at 0 along your reference ray.

Step 03 of 04

Angles add up. If a big angle is split into smaller pieces, the pieces sum to the whole.

If $∠ A B C = 80°$ is split by a ray into $∠ A B D$ and $∠ D B C$ , and $∠ A B D = 35°$ , then $∠ D B C = 80 - 35 = 45°$ .

Step 04 of 04

Triangle interior angles always sum to 180°.

If two angles of a triangle are 50° and 70°, the third is $180 - 50 - 70 = 60°$ .

Key insight

Right = 90°, straight = 180°, full = 360°. Use a protractor: vertex on center, one ray on 0°, read the other ray. Angles split into pieces add up. Triangle interior angles sum to 180°.

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4.J.1 – 4.J.4 · CCSS 4.G.A / 4.MD.C.7 · Full curriculum →