Geometry - Conditional Statements/Proof

Rewire each statement as two if-then statements that are converses of each other.

18. Two angles are supplementary if and only if the sum of their measures is 180.

20. (x-4)(x+6) = 0 if and only if x = 4 or x = -6

22. Theorem 2-5 may also be worded in this way: "If alternate interior angles formed by two lines and a transversal are congruent, then the lines are parallel." Write the converse of this statement. Is the converse true?

Tell whether you think each statement is true or false.
If false, draw a diagram that shows a counterexample.
If true, draw a diagram. List, in terms of he dram, what is given and what is to be proved. Do not write a proof.

24. If a triangle has two congruent sides, then the angles opposite those sides are congruent.

26. Two triangles have equal perimeters only if they have congruent sides.

28. If both pairs of opposite sides of a quadrilateral are paralell, then each side is congruent ot the side opposite it.

30. The diagonals of an equilateral quadrilateral are congruent.

In the conditional statement "if p, then q," p is said to be a sufficient condition for q to occur. Similarly, q is said to be a necessary condition for p. In Exercises 32-37 tell whether the first statement is necessary for the second statement, sufficient for it, or both necessary and sufficient for it.

Eg. First Statement and Second Statement

32. x > 4 and x is positive

34. Line l and m do not intersect.

36. A polygon is equilateral.


38. a) Given: Line AB ll Line DC; Line AD ll Line BC
b) Tell what is given and what is to be proved in the converse of part (a). Then write a proof of the converse.
c) Combine what you have proved in parts (a) and (b) into an if-and-only-if statement.

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    Two angles are supplementary if and only if the sum of their measures is 180.
    1. Two angles are supplementary if the sum of their measures is 180.
    2. The sum of their measures is 180 if two angles are supplementary.

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