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GE.A.1Points, lines, planes01/04

Points, lines, planes, segments, and rays.

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Geometry's vocabulary — five undefined or basic figures that everything else builds on.

Point	→a location, no size — labeled with a capital letter (A)
Line	→extends infinitely in both directions — named by 2 points or a script letter
Plane	→flat surface extending infinitely in all directions — named by 3 non-collinear points
Segment	→part of a line BETWEEN two endpoints — finite length
Ray	→part of a line starting at one endpoint, extending infinitely the other way

GE.A.1 — Points, lines, planes, segments, and rays.

Step 01 of 04

Geometry's vocabulary — five undefined or basic figures that everything else builds on.

Point	→a location, no size — labeled with a capital letter (A)
Line	→extends infinitely in both directions — named by 2 points or a script letter
Plane	→flat surface extending infinitely in all directions — named by 3 non-collinear points
Segment	→part of a line BETWEEN two endpoints — finite length
Ray	→part of a line starting at one endpoint, extending infinitely the other way

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Notation — the bar / arrow on top tells you which one.

$\overline{A B}$	→segment with endpoints A and B (finite)
$A B$	→ray starting at A through B (one direction infinite)
$A B$	→line through A and B (both directions infinite)
$A B$ (no marker)	→the LENGTH of segment AB — a number

The marker matters. $\overline{A B}$ is a figure; $A B$ is a number.

Step 03 of 04

Visualization.

Same letters; the marker changes what's drawn.

Step 04 of 04

Other vocabulary you'll see immediately.

Collinear	→points lying on the SAME line
Coplanar	→points lying in the SAME plane
Midpoint	→the point exactly halfway between two endpoints
Bisect	→cut into two equal parts
Congruent	→same SIZE and SHAPE — symbol ≅
Equal vs. congruent	→numbers are equal (=); figures are congruent (≅)

Key insight

The vocabulary feels heavy because EVERY later skill will lean on it. "AB" and "

\overline{A B}

" are different things; "equal" and "congruent" are different relations. Pay attention to symbols early.

GE.A.2 — Angle pairs — vertical, linear, complementary, supplementary.

Step 01 of 04

When two lines (or two rays) meet, the angles formed have predictable relationships. Knowing four pair types unlocks most "find the missing angle" problems.

Complementary	→two angles that sum to $90°$
Supplementary	→two angles that sum to $180°$
Linear pair	→two adjacent angles whose non-shared sides form a line — automatically supplementary
Vertical angles	→opposite angles formed by two crossing lines — always congruent

Step 02 of 04

Vertical angles are congruent. When two lines cross, the angles opposite each other are equal — no measurement needed.

Pairs labeled with matching arc counts are equal.

$∠1 ≅ ∠3$ and $∠2 ≅ ∠4$ — vertical pairs. $∠1 + ∠2 = 180°$ — they form a linear pair.

Step 03 of 04

Worked example. If $∠1 = (3 x + 10) °$ and $∠3 = (5 x - 30) °$ are vertical angles, find $x$ and the angle measure.

Vertical angles are congruent, so set them equal:

3 x + 10 = 5 x - 30

40 = 2 x \Rightarrow x = 20

∠1 = 3 (20) + 10 = 70°

Step 04 of 04

Complementary vs. supplementary — naming is the hard part.

"Complementary" → 90°	→think C → Corner (a right angle)
"Supplementary" → 180°	→think S → Straight (a straight line)

If two angles are supplementary and one measures $110°$ , the other is $180° - 110° = 70°$ . If complementary and one is $35°$ , the other is $90° - 35° = 55°$ . Pure subtraction.

Key insight

Vertical → equal. Linear pair → sum to 180°. Complementary → sum to 90°, Supplementary → sum to 180°. Half the geometry "find the angle" problems you'll ever see come down to picking the right relationship.

GE.A.3 — Parallel lines cut by a transversal.

Step 01 of 04

When a third line (the transversal) crosses two parallel lines, EIGHT angles are formed — but they fall into just a few equivalence classes. Memorizing the names + which are equal is the whole skill.

Two parallel lines cut by a transversal.

Step 02 of 04

The four equal-angle relationships.

Corresponding	→same position at each intersection — $∠1 ≅ ∠5, ∠2 ≅ ∠6, ∠3 ≅ ∠7, ∠4 ≅ ∠8$
Alternate interior	→between the parallel lines, opposite sides of transversal — $∠3 ≅ ∠6, ∠4 ≅ ∠5$
Alternate exterior	→outside the parallel lines, opposite sides of transversal — $∠1 ≅ ∠8, ∠2 ≅ ∠7$
Vertical (within one intersection)	→ $∠1 ≅ ∠4, ∠2 ≅ ∠3$ etc.

Step 03 of 04

The supplementary pairs. "Same side" pairs sum to $180°$ .

Co-interior (same-side interior)	→ $∠3 + ∠5 = 180°$ and $∠4 + ∠6 = 180°$
Co-exterior (same-side exterior)	→ $∠1 + ∠7 = 180°$ and $∠2 + ∠8 = 180°$
Linear pair (within an intersection)	→ $∠1 + ∠2 = 180°$

Once you know any one of the eight angles, every other is either that measure or its supplement.

Step 04 of 04

Worked example. If $∠3 = 65°$ , find every other angle.

∠3 ∠6 ∠1 ∠8 ∠2 ∠4 = 65° (given) = 65° (alternate interior with ∠3) = 65° (vertical with ∠3) = 65° (alternate exterior with ∠1) = 115° (linear pair with ∠1) = ∠5 = ∠7 = 115° (vertical / corresponding)

Key insight

Two parallel lines + a transversal = only TWO distinct angle measures (some value

x

and its supplement

180° - x

). Every one of the eight angles is one of those two values.

Foundations: Lines & Angles

Points, lines, planes, segments, and rays.

GE.A.1 — Points, lines, planes, segments, and rays.

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GE.A.2 — Angle pairs — vertical, linear, complementary, supplementary.

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GE.A.3 — Parallel lines cut by a transversal.

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Test your understanding: 10 questions, ~10 min.