Unit 1: Foundations of Geometry
The undefined terms and basic objects points, lines, angles that every later definition and proof in the course is built from.
9 topics · 35 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
Points, Lines, and Planes
- How it worksHow do a point, a line, and a plane combine to describe everything in three-dimensional space?
- How it worksHow many points does it take to pin down exactly one plane in space?
- Common mistakeWhy isn't a line in geometry the same as a line segment you can measure with a ruler?
- Going furtherWhat happens to the intersection of two distinct planes that aren't parallel?
Your turn1 of 2
Points A, B, and C do not all lie on the same line. How many distinct planes pass through all three of them?Segments, Length, and Betweenness
- How it worksHow does the segment addition postulate let you add two smaller lengths to get the whole segment?
- Worked exampleFind the length of segment AC if AB is 7 and B lies between A and C with BC equal to 5.
- Common mistakeWhy doesn't a segment that looks longer in a drawing actually have to be longer?
- Going furtherWhat changes about betweenness when three points are graphed on a number line instead of a plane?
Your turn1 of 2
Point B lies between points A and C. AB = 21 and BC = 20. Find AC.Rays and Measuring Angles
- How it worksHow does a protractor turn two rays sharing an endpoint into a number of degrees?
- Worked exampleFind the measure of angle ABC if ray BD splits it into a 25 degree piece and a 40 degree piece.
- Common mistakeWhy isn't a bigger-looking angle necessarily one with a larger degree measure?
- Going furtherWhat happens to an angle's measure when you extend both of its rays further out?
Your turn1 of 2
Ray BD splits angle ABC into two smaller angles measuring 59 degrees and 55 degrees. Find the measure of angle ABC.Classifying Angles by Measure
- How it worksHow do acute, right, obtuse, straight, and reflex angles divide up the entire 360 degree range?
- Worked exampleFind whether an angle of 143 degrees is acute, right, obtuse, straight, or reflex.
- Common mistakeWhy isn't every angle that looks big on the page automatically obtuse?
- Going furtherWhat happens to an angle's classification once it's turned into its reflex counterpart?
Your turn1 of 2
An angle measures 121 degrees. Find the measure of its reflex angle (the angle on the other side of the same two rays).Complementary, Supplementary, and Linear Pairs
- How it worksHow does a linear pair sitting on a straight line force its two angles to sum to 180 degrees?
- Worked exampleFind the measure of an angle that is complementary to a 37 degree angle.
- Worked exampleFind both angles in a supplementary pair where one angle is three times the other.
- Common mistakeWhy aren't two angles automatically supplementary just because they look like they add up to a straight line?
Your turn1 of 2
Angle A and angle B are complementary. Angle A measures 61 degrees. Find the measure of angle B.Vertical Angles
- How it worksHow does the linear pair relationship prove that vertical angles must be congruent?
- Worked exampleFind the measures of all four angles formed when two lines cross and one angle measures 55 degrees.
- Going furtherWhat happens to the vertical angle pairs when the crossing lines are rotated to be perpendicular?
Your turn1 of 2
Two lines cross, forming four angles. One of the angles measures 121 degrees. Find the measure of the angle adjacent to it (its linear-pair partner along one of the lines).Angle Bisectors and Segment Bisectors
- How it worksHow does a bisector ray split one angle into two congruent angles?
- Worked exampleFind the two angle measures created when a 78 degree angle is bisected.
- Common mistakeWhy doesn't a segment bisector have to cross its segment at a right angle?
- Ties togetherHow does a perpendicular bisector combine the ideas of a segment bisector and a right angle at once?
Your turn1 of 2
Ray BD bisects angle ABC, which measures 127 degrees. Find the measure of each of the two resulting angles.Conditional Statements and Their Converses
- How it worksHow do you build the converse, inverse, and contrapositive of a single conditional statement?
- Worked exampleFind the converse of the statement 'if a shape is a square, then it has four right angles' and decide if it is true.
- Common mistakeWhy isn't the converse of a true statement automatically true too?
- Going furtherWhat happens when a conditional statement and its converse are both true at once?
Your turn1 of 2
Consider the true conditional: "If a shape is a square, then it has four right angles." What is its converse, and is the converse true?Inductive vs. Deductive Reasoning
- How it worksHow does deductive reasoning chain definitions and postulates together to guarantee a conclusion?
- How it worksHow does inductive reasoning use a pattern from specific examples to guess a general rule?
- Common mistakeWhy doesn't noticing a pattern in a few examples count as a mathematical proof?
- Going furtherWhat happens to a conjecture found by inductive reasoning once a single counterexample shows up?
Your turn1 of 2
A student checks that the sum of the first 5 odd numbers is always a perfect square (1, 1+3=4, 1+3+5=9, ...) and concludes this pattern is a proven mathematical fact. What kind of reasoning is this, and is the conclusion justified?