Point, line and plane are undefined terms; everything else is built from them by definitions. A postulate is accepted without proof; a theorem is proved. A proof is a chain of statements, each with a reason.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Try it first
Two lines cross, making four angles. Angles 1 and 3 are opposite each other where the two lines cross; angle 2 sits between them, beside ∠1 on one straight line and beside ∠3 on the other. ∠1 = 65°. Using only that angles on a straight line add to 180°, what is ∠3, in degrees?
degrees
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
Undefined terms
Point, line and plane are not defined, because every definition needs earlier words, and something has to come first. They are described instead: a point has position but no size, a line extends without end, a plane is a flat surface without edge.
Definitions
A definition uses the undefined terms to name something new, and it works in both directions. Perpendicular lines meet at 90°, and lines that meet at 90° are perpendicular. A midpoint divides a segment into two equal parts, and equal parts mean a midpoint.
Postulates
Some statements are accepted without proof, because proof has to start somewhere. Through any two points there is exactly one line. Angle measures add: ∠AOB + ∠BOC = ∠AOC when B lies between. Angles on a straight line add to 180°.
A theorem is earned
A theorem is proved from definitions, postulates and theorems already established. "Vertical angles are equal" is a theorem: it follows from angles on a straight line adding to 180°, as Try It Yourself shows.
What a proof is
Each statement follows from what came before, with a reason given for every step. The reasons are definitions, postulates, earlier theorems and the properties of equality. "It looks right" is never one of them.
Properties of equality are reasons too
Subtracting 40 from both sides is justified by the subtraction property of equality; replacing a quantity with an equal one is substitution. Algebra inside a proof carries a reason like everything else.
Step 2: Try It Yourself
Tap and try it out.
Prove: vertical angles are equal
- StatementReason
- 1∠1 + ∠2 = 180°angles on a straight line add to 180°
Step 0 of 3
60° is an acute angle.
Step 3: In Real Life
A courtroom
A lawyer starts from agreed facts and reasons to a verdict step by step. Geometry does the same: definitions and postulates are the agreed facts, and a proof is the argument.
Step 4: Watch an Example
One step at a time.
Watch Priya Prove ∠BOC = 55°
Ray OB lies inside ∠AOC. ∠AOB = 35° and ∠AOC = 90°. Priya proves the rest of the angle is 55°, with a reason on every line.
- Step 1
The given facts go down first, each with the reason "given": ∠AOB = 35° and ∠AOC = 90°. A proof may use only what it has been given or can justify, so the givens are written before anything else.
Watch Dev Find x, With a Reason on Every Line
Ray OB lies inside ∠AOC. ∠AOB = 2x, ∠BOC = x + 12 and ∠AOC = 90°. Dev writes the first two lines; you supply the next three.
- Step 1
The givens first: ∠AOB = 2x, ∠BOC = x + 12 and ∠AOC = 90°, each with the reason "given". Letters in the givens change nothing about the shape of the proof.
Step 5: Your Turn
Practice makes it stick.
The Pizza Slice
Problem 1 of 2
A pizza shop sells by the angle: the whole 360° pizza is $36, so every degree costs $0.10. Ray OB lies inside ∠AOC, a 90° quarter of the pizza, and ∠AOB = 40°. The slice BOC is sold on its own. What does it cost, in dollars?
dollars
The Frame Corner
Problem 2 of 2
A framer cuts two boards to meet at a square 90° corner, so the two cut angles are complementary: together they make the 90°. One cut is 37°. The shop charges $2.50 for every degree a cut is set away from the standard 45°, on each cut. What does the corner cost, in dollars?
dollars
Reasons and Angles
Each answer has a short proof behind it. Name the reason as you go.
1 of 5
Two angles are complementary and one is 34°. What is the other, in degrees?
degrees
2 of 5
Two angles are supplementary and one is 103°. What is the other, in degrees?
degrees
3 of 5
Ray OB lies inside ∠AOC. ∠AOB = 48° and ∠AOC = 131°. What is ∠BOC, in degrees?
degrees
4 of 5
Two lines cross. ∠1 = 71°, and ∠3 is the angle opposite it across the crossing. What is ∠3, in degrees?
degrees
5 of 5
Ray OB lies inside ∠AOC. ∠AOB = 3x, ∠BOC = 2x and ∠AOC = 90°. What is x?
Whose Step?
One line in each proof is a slip: a statement that does not follow, or a reason that is not a reason. Find it, then fix it.
1 of 6
Ray OB lies inside ∠AOC. ∠AOB = 42° and ∠AOC = 90°. Maya proved ∠BOC = 138°. One line is a slip. Which?
Maya's proof
- 1∠AOB = 42°, ∠AOC = 90°
- 2∠AOB + ∠BOC = ∠AOC
- 342 + ∠BOC = 180
- Answer∠BOC = 138− 42
2 of 6
Fix Maya's line 3 and finish the proof. What is ∠BOC, in degrees?
degrees
3 of 6
Two lines cross, and ∠1 = 58°. Leo proved that ∠3, opposite ∠1, is 58°. Every statement is true, but one reason is not a reason. Which line? The reasons Leo wrote are printed with the options below.
Leo's proof
- 1∠1 = 58°
- 2∠1 = ∠3
- Answer∠3 = 58°
4 of 6
What reason should Leo write beside ∠1 = ∠3?
5 of 6
∠1 and ∠2 are supplementary, and ∠1 = 64°. Ana proved ∠2 = 26°. One line is a slip. Which?
Ana's proof
- 1∠1 = 64°
- 2∠1 + ∠2 = 90
- 364 + ∠2 = 90
- Answer∠2 = 26− 64
6 of 6
Fix Ana's line 2 and finish the proof. What is ∠2, in degrees?
degrees
Pick the Move
Decide the first line after the givens, and its reason, before writing anything else.
1 of 2
∠1 and ∠2 are supplementary. ∠1 = 2x and ∠2 = x + 30. After the givens, which line comes next?
2 of 2
Ray OB lies inside ∠AOC. ∠AOB = 3x, ∠BOC = x + 8 and ∠AOC = 60°. After the givens, which line comes next?
Same Kind?
Definition, postulate or theorem — each statement is one of the three. Select the ones the question asks for.
1 of 2
Select every statement that is a postulate: accepted without proof, and a place a chain of reasoning can start.
2 of 2
Select every statement that is a definition, and so works in both directions.
Step 6: Quick Check
Show what you know.
Question 1 of 2
Supplementary to 118°?
Question 2 of 2
What must every line of a proof have?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
Ray OB lies inside ∠AOC and splits it into two equal parts: ∠AOB = ∠BOC. ∠AOB = 3x − 5 and ∠AOC = 4x + 10. What is x?
Stretch 2 of 2
∠1 and ∠2 are supplementary. ∠1 = 5x + k and ∠2 = 7x − k, where k is a number nobody has told you. Sam says he can find x anyway. What is x?
What You Learned
- Point, line and plane are undefined terms; definitions build everything else and work both ways.
- A postulate is accepted; a theorem is proved.
- Every line of a proof carries a reason: a definition, a postulate, a theorem, or a property of equality.
For the grown-up
The misconception this lesson targets is that a diagram is a reason: "the angles look equal", or a protractor reading, offered to justify a line. A proof accepts only definitions, postulates, theorems already proved and the properties of equality; the diagram shows what to prove, never why. Its cousin is the 90 and 180 swap, complementary written where supplementary was meant, which a quoted definition catches and a remembered one does not.