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.
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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.
false
18. If two angles are supplementary, then the sum of their measures is 180.
If the sum of two angles' measures is 180, then the angles are supplementary.
20. If (x-4)(x+6) = 0, then x = 4 or x = -6.
If x = 4 or x = -6, then (x-4)(x+6) = 0.
22. Converse: If the lines are parallel, then alternate interior angles formed by two lines and a transversal are congruent.
The converse is true.
24. True. (Diagram: Triangle ABC with AB = AC) Given: Triangle ABC has two congruent sides (AB = AC). To be proved: The angles opposite the congruent sides (angle B and angle C) are congruent.
26. True. (Diagram: Two triangles with equal perimeters) Given: Two triangles have equal perimeters. To be proved: The two triangles have congruent sides.
28. True. (Diagram: Quadrilateral with both pairs of opposite sides parallel) Given: Both pairs of opposite sides of a quadrilateral are parallel. To be proved: Each side is congruent to the side opposite it.
30. True. (Diagram: Equilateral quadrilateral with congruent diagonals) Given: The diagonals of an equilateral quadrilateral are congruent. To be proved: None (no additional statement is needed).
32. Both necessary and sufficient. (First Statement: x > 4, Second Statement: x is positive)
34. Both necessary and sufficient. (First Statement: Line l and m do not intersect, Second Statement: Line l and m are parallel)
36. First Statement is necessary. (First Statement: A polygon is equilateral, Second Statement: The polygon has all its sides congruent)
38. a) Given: Line AB is parallel to Line DC, Line AD is parallel to Line BC.
b) Converse: Given: Line AB is parallel to Line DC, Line AD is parallel to Line BC. Prove: Line AD is parallel to Line BC, Line AB is parallel to Line DC.
c) If Line AB is parallel to Line DC and Line AD is parallel to Line BC, then Line AD is parallel to Line BC and Line AB is parallel to Line DC.
18. The converse of the statement "Two angles are supplementary if and only if the sum of their measures is 180" would be "If the sum of the measures of two angles is 180, then the angles are supplementary." To determine if the converse is true, we need to check if all angles with a sum of 180 are necessarily supplementary.
20. The converse of the equation "(x-4)(x+6) = 0 if and only if x = 4 or x = -6" would be "If x = 4 or x = -6, then (x-4)(x+6) = 0." To determine if the converse is true, we need to check if plugging in x = 4 or x = -6 into the equation results in (x-4)(x+6) equaling zero.
22. The converse of the statement "If alternate interior angles formed by two lines and a transversal are congruent, then the lines are parallel" would be "If the lines are parallel, then the alternate interior angles formed by two lines and a transversal are congruent." To determine if the converse is true, we need to check if all pairs of lines that are parallel necessarily have congruent alternate interior angles.
24. This statement is true. In a diagram, we can draw a triangle with two congruent sides and label the angles opposite those sides as A and B. To prove that the angles are congruent, we can use the Isosceles Triangle Theorem which states that in an isosceles triangle, the angles opposite the congruent sides are congruent.
26. This statement is false. In a diagram, we can draw two triangles with different side lengths but the same perimeter. This counterexample shows that equal perimeters do not necessarily imply congruent sides.
28. This statement is true. In a diagram, we can draw a quadrilateral with both pairs of opposite sides parallel and label the sides as AB, BC, CD, and DA. To prove that each side is congruent to the side opposite it, we can use the definition of a parallelogram which states that opposite sides of a parallelogram are congruent.
30. This statement is true. In a diagram, we can draw an equilateral quadrilateral and label the diagonals as AC and BD. The congruence of the diagonals can be proven using the properties of an equilateral quadrilateral.
32. The first statement ("x > 4") is a necessary condition for the second statement ("x is positive") since if x is not greater than 4, it cannot be positive. However, the first statement is not sufficient for the second statement since there are values of x greater than 4 that can be negative.
34. The first statement ("Line l and m do not intersect") is both a necessary and sufficient condition for the second statement ("Line l is parallel to line m") since if the lines do not intersect, they are parallel, and if they are parallel, they do not intersect.
36. The first statement ("A polygon is equilateral") is a necessary and sufficient condition for the second statement ("A polygon has congruent sides") since an equilateral polygon by definition has all sides congruent, and if a polygon has congruent sides, it is equilateral.
38.
a) Given: Line AB ll Line DC; Line AD ll Line BC
b) In the converse of part (a), we need to prove that if Line AD is parallel to Line BC and Line AB is parallel to Line DC, then Line AB is parallel to Line DC and Line AD is parallel to Line BC. To write a proof of the converse, we can use the transitive property of parallel lines which states that if Line AB is parallel to Line DC and Line DC is parallel to Line AD, then Line AB is parallel to Line AD.
c) Combining the proof from part (a) that if Line AB is parallel to Line DC and Line AD is parallel to Line BC, then Line AB is parallel to Line DC and Line AD is parallel to Line BC with the proof from part (b) that if Line AD is parallel to Line BC and Line AB is parallel to Line DC, then Line AB is parallel to Line DC and Line AD is parallel to Line BC, we can conclude that "Line AB is parallel to Line DC and Line AD is parallel to Line BC if and only if Line AB is parallel to Line DC and Line AD is parallel to Line BC."