☰ Contents · Geometry

Proof problems

Lessons 57 · 1 lessons · Geometry Grade 7, corrected and expanded 3rd edition. Yangiyo‘l Poligraf Servis, Tashkent, 2017 (approved by the Ministry of Public Education)
57

Proof problems

Textbook: pp. 148–149
GoalBe able to write proof problems as given, to prove and reasons.
New words
proof problem · isbotlashga doir masalajustification · asoslashbisector · bissektrisamedian · mediana
Explanation

A proof problem is like a small theorem: you must show that a given statement is true. First write “Given” and “To prove” and draw a diagram. Then justify every step by a definition, an axiom or a theorem already proved. Often the triangle congruence criteria (SAS, ASA, SSS) are used, and equality of corresponding elements is concluded. Useful tricks: draw an auxiliary line or label angles with letters. At the end restate what was proved. That something looks true on the diagram is not a proof.

Worked examples
Lines AB and CD meet at O, and OM bisects ∠AOC. Prove that the extension of OM (ray ON) bisects ∠BOD. ∠MOA = ∠MOC = α. ∠NOB and ∠MOA are vertical, so ∠NOB = α; ∠NOD and ∠MOC are also vertical, so ∠NOD = α. Hence ON is a bisector.
In isosceles triangle ABC with AB = AC, M and N are the midpoints of AB and AC. Prove BN = CM. △ABN and △ACM: AB = AC (given), AN = AM (halves of equal sides), ∠A is common. By SAS △ABN = △ACM, so corresponding sides are equal: BN = CM.
Class activity

Play “proof” with a friend: one writes a step, the other names its reason (theorem or definition).

Practice
1
One of two vertical angles is 84°. How many degrees is each part made by its bisector?
2
In isosceles ABC (AB = AC) with M, N the midpoints of the legs, which triangles are compared to prove BN = CM?
3
AB = AC = 10 cm, and M, N are the midpoints of the legs. How many cm is AM?
4
Why is a reason written for every step of a proof?