☰ Contents · Geometry

Converse theorem. Parallel lines and angles

Lessons 33–34 · 2 lessons · Geometry Grade 7, corrected and expanded 3rd edition. Yangiyo‘l Poligraf Servis, Tashkent, 2017 (approved by the Ministry of Public Education)
33

The converse theorem

Textbook: pp. 86–87
GoalKnow and form a converse theorem.
New words
converse theorem · teskari teoremahypothesis and conclusion · shart va xulosadirect theorem · to‘g‘ri teoremafalse statement · noto‘g‘ri tasdiq
Explanation

If the hypothesis and conclusion of a theorem are swapped, a new statement appears; if it is proved, it is called the converse theorem of the given one. Direct theorem: if A, then B (A ⇒ B). Converse: if B, then A (B ⇒ A). For example, the converse of “the base angles of an isosceles triangle are equal” is “a triangle with two equal angles is isosceles”; it is also true and is a test for an isosceles triangle. But a converse is not always true: the converse of “vertical angles are equal” is “equal angles are vertical”, which is false because equal angles need not be vertical. One example (a counterexample) is enough to show a statement is false.

Worked examples
Direct: “An equilateral triangle is isosceles.” Converse: “An isosceles triangle is equilateral” — false (counterexample: sides 5, 5, 8).
Direct: “If it rains, there are clouds.” Converse: “If there are clouds, it rains” — not always true.
Class activity

“Say the converse”: pupils take turns stating a theorem; partners form its converse and discuss whether it is true.

Practice
1
Write the converse of “If a triangle is isosceles, its base angles are equal”.
2
Write the converse of “If a ∥ b, the alternate interior angles are equal”.
3
How do we show “equal angles are vertical” is false?
4
Why can a converse not always be assumed true?