Geometry has its own punctuation. Reading it correctly is half the work.
Read the notation
What is the difference between AB with a bar over it and AB with an arrow over it?
A bar means the segment — the piece between A and B, with both ends. An arrow means a ray, starting at A and continuing past B forever. Plain AB with no mark means the length, which is a number. Three marks, three different things, and questions rely on the distinction.
1
The three undefined terms.
Term
Think of
Named by
Point
a position, no size
one capital letter, A
Line
straight, endless both ways
two points on it, or one lower-case letter
Plane
a flat surface, endless
three points not in a line
They are called undefined because every attempt to define them uses the others. Geometry starts by agreeing on these and building everything else from them.
Fact or word
What it says
Two points
determine exactly one line
Three points not all on one line
determine exactly one plane
Two lines that cross
meet in one point
Two different planes that cross
meet in a line
Collinear / coplanar
on the same line / in the same plane
Parallel lines
in the same plane and never meet
Perpendicular lines
meet at a 90° angle
2
Segment, ray, line — how far each one goes.
A segment has two endpoints, so it has a definite length. A ray has one endpoint and a line has none, so neither can be measured.
Ray order matters
Ray AB starts at A. Ray BA starts at B and points the other way. They are
different rays, even though segment AB and segment BA are the
same segment.
3
Betweenness and the segment addition postulate.
If B is between A and C, then AB + BC = AC.
AB = 3x, BC = x + 4, AC = 28
B between A and C
3x + (x + 4) = 28
part + part = whole
4x = 24, x = 6
solve
AB = 18, BC = 10
and 18 + 10 = 28 ✓
The same part-plus-part-equals-whole idea covers angles: if a ray sits inside an
angle, the two pieces add to the whole.
4
Angles, named and classified.
Type
Measure
Acute
between 0° and 90°
Right
exactly 90°
Obtuse
between 90° and 180°
Straight
exactly 180°
Reflex
between 180° and 360°
The vertex goes in the middle
∠ABC has its vertex at B, the middle letter. Writing ∠BAC names a
completely different angle. When only one angle sits at a vertex you may shorten it
to ∠B — but not when several do.
5
Midpoints, bisectors, and congruence marks.
Midpoint of a segmentsplits it into two equal halvesAM = MB
Angle bisectorsplits an angle in twoequal halves
Perpendicular bisectorcuts in half AND at 90°both conditions
Tick marks in a figurematching marks mean equalgiven, not assumed
Read a diagram's marks, not its appearance. Two segments that look equal are only equal if the marks say so — and unmarked right angles may not be right angles at all.
Remember the flow
Bar = segment, arrow = ray, nothing = length · Vertex in the middle · Part + part = whole · Trust the marks, not the picture
Ray AB and ray BA point in opposite directions. Segment AB and segment BA do not.
6
Halves with algebra: set them equal, solve, then answer the question.
A midpoint or a bisector gives two EQUAL parts. Equal parts make an equation.
Ray BD bisects ∠ABC. ∠ABD = (2x + 5)° and ∠DBC = (4x − 15)°. Find ∠ABC.
2x + 5 = 4x − 15
a bisector makes the halves equal
20 = 2x, so x = 10
solve
∠ABD = 2(10) + 5 = 25°
check: 4(10) − 15 = 25 ✓
∠ABC = 2 × 25° = 50°
the whole angle was asked
Three numbers, one answer
This problem produces 10, 25 and 50. Stopping at x = 10 or at one half, 25°, answers a question nobody asked. Before you write the answer, reread what the question names: x, a half, or the whole.
7
Variations: x in the whole, a straight angle, a number line with negatives.
Part + part = whole and equal halves, in three new settings.
B is between A and C; AB = x + 2, BC = 2x − 1, AC = 4x − 5. Find AC.
(x + 2) + (2x − 1) = 4x − 5
part + part = whole
3x + 1 = 4x − 5, so x = 6
x on both sides
AC = 4(6) − 5 = 19
check: 8 + 11 = 19 ✓
When the whole also contains x, collect the x terms on one side as usual.
∠PQR is a straight angle split by ray QS into (3x)° and (x + 20)°.
3x + x + 20 = 180
a straight angle is 180°
4x = 160, so x = 40
solve
120° and 60°
adding to 180 ✓
The whole is never written down here; the word "straight" supplies it.
On a number line A is at −7 and B is at 5. Find AB and the midpoint M.
AB = 5 − (−7) = 12
right minus left
M = −7 + 52 = −1
average the coordinates
AM = 6, MB = 6
equal halves ✓
A length is never negative; a coordinate can be. Subtracting a negative adds.
On the exam
Expect notation (segment, ray, line, which letter is the vertex), the vocabulary words (collinear, coplanar, parallel, perpendicular), and short algebra on a midpoint, a bisector or segment addition. The fastest route for the algebra is one equation — equal halves, or part + part = whole — then substitute x back. The 10-second check: the parts you found must add to the whole, and you must be answering for x, one part or the whole exactly as asked.
Now you try
Practice problem
Find the length
Point M is the midpoint of segment PQ. If PM = 4x − 3 and MQ = 2x + 9, how long is PQ?
A midpoint makes the halves equal: 4x − 3 = 2x + 9, so 2x = 12 and x = 6. Each half is 4(6) − 3 = 21, so the whole segment PQ is 2 × 21 = 42. A is x, and B is one half — the question asked for the whole segment.
Hint: three steps — set the halves equal, solve for x, then double.
Stopping after the second step answers a different question.
