Cogito
Integrated Math 2 · Chapter 7 · Lesson 3
Writing a Proof
A chain of justified steps.
12 problems · about 22 minutes · G-CO.C.10, G-SRT.B.4
What this lesson teaches
The student writes short two-column and paragraph proofs with justified steps.
- A proof runs from given information to a conclusion, justifying every step.
- Reasons are given facts, definitions, or established theorems.
- Plan backwards from the conclusion, then write forwards.
Warm Up
Straightforward practice. Get the method working first.
5 problemsThree pairs of equal sides. Which criterion? 1 SSS, 2 SAS, 3 ASA.
Answer 1
Why SSS.
Why can a diagram not justify a step?
Answer Appearance is not evidence; only stated or derived facts count.
Why A proof rests on facts, not on how the picture looks.
Is "the diagram looks that way" a valid reason? 1 yes, 0 no.
Answer 0
Why Only stated or derived facts.
Is a previously proved theorem a valid reason? 1 yes, 0 no.
Answer 1
Why Established results may be used.
Two triangles with two equal sides and an equal included angle. Which criterion? 1 SSS, 2 SAS, 3 ASA.
Answer 2
Why Side, angle, side.
Build It Up
The same ideas with more to keep track of.
3 problemsHow many statements in a two-column proof need a reason?
Answer 1
Why Enter 1 for every one of them.
A triangle has angles 70° and 60°. Third angle, in degrees?
Answer 50
Why 180 − 130.
Once two triangles are congruent, are corresponding parts equal? 1 yes, 0 no.
Answer 1
Why That is what congruence gives.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the proof-writing process in order.
Answer 1. Read the given information and the conclusion wanted. 2. Work backwards to find what would give that conclusion. 3. Write the statements forwards from the given facts. 4. Attach a reason to every statement.
Why Plan backwards, write forwards.
Two angles and the included side equal. Which criterion? 1 SSS, 2 SAS, 3 ASA.
Answer 3
Why Angle, side, angle.
The Shared Side: A side shared by two triangles is equal to itself. How many pairs of equal sides does that supply?
Answer 1
Why 1.
The Criterion: Three pairs of equal sides. Which criterion? 1 SSS, 2 SAS, 3 ASA.
Answer 1
Why SSS.