Definitions, Postulates & Reasoning — Challenge
Geometry runs on agreed definitions. Once a word is pinned down, arguments about it stop.
What these problems 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
How many degrees are there in a right angle?
- 45
- 60
- 90
- 180
- 360
Worked answer
90
A right angle is a quarter of a full turn.
A full turn is 360 degrees, so a quarter of it is 90 degrees.
Problem 2
What do the three angles of a triangle add up to, in degrees?
- 90
- 180
- 270
- 360
- 540
Worked answer
180
Tearing off the three corners of any triangle and placing them together makes a straight line.
A straight line is 180 degrees, so the three angles add to 180.
This holds for every triangle, whatever its shape.
Problem 3
An angle measures 120 degrees. What kind of angle is it?
- Acute
- Right
- Obtuse
- Straight
- Reflex
Worked answer
Obtuse
An acute angle is under 90 degrees and a right angle is exactly 90.
An obtuse angle is between 90 and 180 degrees, and 120 falls in that range.
A reflex angle would be more than 180 degrees.
Problem 4
How many sides does a pentagon have?
- 4
- 5
- 6
- 7
- 8
Worked answer
5
The prefix penta means five.
So a pentagon has five sides, and five angles to match.
Problem 5
Two lines in the same plane never meet. What are they called?
- Perpendicular
- Parallel
- Intersecting
- Skew
- Congruent
Worked answer
Parallel
Lines in a plane that never meet, however far they are extended, are parallel.
Perpendicular lines do meet, at a right angle.
Skew lines also never meet, but they must lie in different planes, which is ruled out here.
Problem 6
How many degrees are there in a straight line?
- 45
- 180
- 270
- 360
- 90
Worked answer
180
A straight line is half a full turn.
Half of 360 degrees is 180 degrees.
Problem 7
An angle measures 45 degrees. What kind of angle is it?
- Acute
- Right
- Obtuse
- Straight
- Reflex
Worked answer
Acute
An angle smaller than 90 degrees is acute.
45 is less than 90, so the angle is acute. It is exactly half a right angle.
Problem 8
At how many degrees do perpendicular lines meet?
- 30
- 45
- 60
- 90
- 180
Worked answer
90
Perpendicular means meeting at a right angle.
A right angle is 90 degrees.