Definitions, Postulates & Reasoning — Practice
Geometry runs on agreed definitions. Once a word is pinned down, arguments about it stop.
What these questions cover
These questions use 9 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Bisect — Cut exactly in half.
- Congruent — Same size and same shape. Written △ ABC ≅ △ DEF.
- Midpoint — The point halfway between two others: average the x values and average the y values.
- Undefined term — Point, line and plane. Described but never defined, because any definition would use them.
- Postulate — A statement accepted without proof, such as "two points determine exactly one line."
- Segment — Two endpoints and everything between them. Written AB.
- Ray — One endpoint and everything beyond it in one direction. Written AB.
- Counterexample — A single case that makes a claim false. One is enough.
- Number line — A line with numbers increasing to the right.
Problem 1
Write the symbol for the segment with endpoints A and B.
Worked answer
AB with a bar
AB
The bar means a segment, not a length.
Problem 2
Does AB stop at B? Answer yes or no.
Worked answer
no
A ray starts at A and continues past B forever.
So it does not stop.
Problem 3
B is between A and C. AB = 7 and BC = 12. Find AC.
Worked answer
19
AB + BC = AC
7 + 12 = 19
Problem 4
B is between A and C. AC = 30 and AB = 11. Find BC.
Worked answer
19
AB + BC = AC
BC = 30 - 11 = 19
Problem 5
M is the midpoint of PQ and PM = 9. Find PQ.
Worked answer
18
MQ = PM = 9
PQ = 9 + 9 = 18
Problem 6
B is between A and C. AB = 3x + 2, BC = 2x - 1, AC = 26. Find x.
Worked answer
5
(3x+2)+(2x-1) = 26
5x + 1 = 26
x = 5
Problem 7
M is the midpoint of PQ. PM = 4y - 3 and MQ = y + 9. Find PQ.
Worked answer
26
4y - 3 = y + 9, so y = 4
PM = 13
PQ = 26
Problem 8
OB bisects ∠ AOC. If m∠ AOB = 34°, find m∠ AOC.
Worked answer
68
m∠ AOC = 2 × 34
= 68°
Problem 9
OB is inside ∠ AOC. m∠ AOB = 28° and m∠ AOC = 91°. Find m∠ BOC.
Worked answer
63
28 + m∠ BOC = 91
m∠ BOC = 63°
Problem 10
Which of these is an undefined term in geometry?
- Point
- Segment
- Angle
- Midpoint
Worked answer
Point
Point, line and plane are undefined.
Segment, angle and midpoint are all defined using them.
Problem 11
"Through any two points there is exactly one line" is a
- postulate
- theorem
- definition
- counterexample
Worked answer
postulate
It is assumed without proof.
That makes it a postulate.
Problem 12
Is AB = CD written correctly? Answer yes or no.
Worked answer
no
Segments use ≅, numbers use =.
It should be AB ≅ CD or AB = CD.
Problem 13
OB bisects ∠ AOC. m∠ AOC = (6x+4)° and m∠ AOB = (2x+14)°. Find m∠ AOC.
Worked answer
76
2(2x+14) = 6x+4
4x+28 = 6x+4, so x = 12
6(12)+4 = 76°
Problem 14
P, Q, R are collinear with Q between. PQ = 2x, QR = x + 5, PR = 35. Find QR.
Worked answer
15
2x + x + 5 = 35
3x = 30, so x = 10
QR = 10 + 5 = 15
Problem 15
M is the midpoint of AB. A is at -4 and M is at 3 on a number line. Where is B?
Worked answer
10
M - A = 3 - (-4) = 7
B = 3 + 7 = 10
Problem 16
Three rays from O make angles m∠ AOB = (x)°, m∠ BOC = (2x)°, and m∠ AOC = 72°. Find m∠ BOC.
Worked answer
48
x + 2x = 72
x = 24
m∠ BOC = 48°