Review of Prerequisite Skills
Section 2.3 The Converse of a Conditional Statement
The theorems we have accepted and the results we have proved from them are examples of conditional statements. If one statement (the premise) is true, then another statement (the conclusion) is true. Symbolically, if p is the premise and q is the conclusion, we write p → q, which is read, “p implies q,” or “if p then q.”
An idea closely related to a conditional statement p → q is that of its converse, which we write as q → p. We cannot conclude that the converse is true because the original statement is true. Consider the statement, If the traffic light is red then we stop the car. This statement is true for both legal and safety reasons. The con-verse is If we stop the car, then the light is red. This statement is not necessarily true; we can stop the car for a variety of reasons.
If a statement is true (p → q) and its converse is also true (q → p), we write p ↔ q, which is read, “p if and only if q (or q if and only if p).” We sometimes write this as “p iff q.”
When we write p↔ q we recognize that the truth of either of the statements depends upon the truth of the other. This type of statement is said to be
biconditional.
EXAMPLE 1 For each of the following statements a. state the converse
b. determine whether the converse is a true statement
c. if the converse is true, restate the sentence as an …if and only if….
statement
1. If one side of a balance falls, there is more weight on that side than on the other.
2. If one of two integers is even and the other is odd, then the sum of the integers is odd.
3. If it is spring, then the grass is green.
Solution
1. The converse of this statement is, If there is more weight on one side of a bal-ance than on the other, then one side of the balbal-ance falls. This is certainly true.
The biconditional statement is, One side of a balance falls if and only if there is more weight on that side than on the other.
2. The converse of this statement is, If the sum of two integers is odd, then one of the integers is even and one is odd. This is certainly true. The biconditional statement is, The sum of two integers is odd if and only if one of them is even and the other is odd.
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3. The converse of this statement is, If the grass is green, then it is spring. This statement is certainly not true.
EXAMPLE 2 Is it true that p ↔ q, for these two statements?
p: Two angles are vertically opposite. q: The two angles are equal.
Solution
Part 1 p → q
The first statement we will prove is, If two angles are vertically opposite then they are equal.
Since ∠COA ∠COB 180º, ∠COB 180º ∠COA Since ∠COA ∠AOD 180º, ∠AOD 180º ∠COA Therefore ∠COB ∠AOD
The statement is true.
Part 1 q → p
The second statement is, If two angles are equal then they are vertically opposite angles.
We can demonstrate that this statement is not true by constructing an example showing it to be false.
In isosceles triangle ABC,∠ABC ∠ACB but these two angles are not vertically opposite each other. Then p ↔ q is not true because p → q and q → p are not both true.
THEOREM Prove the biconditional statement, A point is on the right bisector of a given line segment if and only if it is equidistant from the ends of the segment.
Solution
First we prove that any point on the right bisector of a line segment is equidistant from the end points of the line segment.
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Proof
Let AB be any line segment and XY be its right bisector, cutting AB at D. Let P be any point on XY. Join P to A and P to B.
In triangles PAD and PBD, AD BD (right bisector)
∠PDA ∠PDB (right angles)
PD PD (same line)
Then ∆PAD ∆PBD (side-angle-side)
Then PA PB
Now we prove that if a point is equidistant from the endpoints of a line segment, then it is on the right bisector of the line segment.
Proof
We are given PA PB. We join P to M, the midpoint of AB.
In triangles PAM and PBM,
PA PB (given)
AM BM (constructed)
PM PM (same line segment)
Then ∆PAM ∆PBM (side-side-side)
Then ∠PMA ∠PMB and, since their sum is 180º,
∠PMA ∠PMB 90º Then PM is the right bisector of AB.
We combine the results of these two properties in the Right Bisector Theorem.
EXAMPLE 3 Determine the position of a point P equidistant from three given points A, B, and C that are not in a straight line.
Solution
If P is equidistant from A and B it must lie on the right bisector of AB, so P is a point on DE, the right
bisector of AB.
Similarly, P is a point on FG, the right bisector of BC.
The point P is equidistant from A, B, and C, and so PA PB PC.
DY
A B
P X
A M B
P
Right Bisector Theorem A point is on the right bisector of a given line segment if and only if it is equidistant from the end points of the line segment.
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A
THEOREM A point is on the bisector of an angle if and only if it is equidistant from the arms of the angle.
Proof Part 1
If a point is on the bisector of an angle, it is equidistant from the arms of the angle.
Let BD be the bisector of ∠ABC and let P be any point on BD.
From P draw perpendicular lines to meet the sides BA and BC at X and Y respec-tively. We will prove that PX PY.
In ∆PXB and ∆PYB,
∠PBX ∠PBY (given)
PB PB (same line)
∠PXB ∠PYB 90º (construction)
Therefore ∆PBX ∆PBY (angle-angle-side)
Then PXPY and P is equidistant from the arms of the angle.
Part 2
If a point is equidistant from the angle arms, then it is on the bisector of the angle.
Let P be any point on line BD such that perpendic-ulars PX and PY are equal.
In ∆PBX and ∆PBY
PX PY (given)
PB is common
∠PXB ∠PYB 90º (given)
Then ∆PXB ∆PBY (hypotenuse-side)
Then ∠PBX ∠PBY and BP is the bisector of the angle.
B Y C
D A X
P xx
B Y C
D A X
P
Angle Bisector TheoremA point is on the bisector of an angle if and only if it is equidistant from the sides of the angle.
Part A
1. State the converse of each of the following statements.
a. If a triangle has three unequal sides, then it has three unequal angles.
b. If it rains, then we will get wet.
c. If a four-sided figure has four equal angles, then it is a square.
d. If the fruit is yellow, then it is a banana.
e. If today is Saturday, then it is the weekend.
f. If all answers on a test are incorrect, then several errors have been made.
g. If an integer is a prime, then it is not divisible by 2, 3, 5, or 7.
2. For each of the statements in Question 1, determine whether a. the statement is true
b. the converse is true
c. the statement and its converse form a biconditional statement 3. Determine which of the following statements are true.
a. A positive integer is prime if and only if it is odd.
b. An integer is divisible by 2 if and only if it is even.
c. Figures are congruent if and only if they are similar.
d. A four-sided figure is a parallelogram if and only if it is a rectangle.
e. An integer is divisible by 5 if and only if it ends in a 5.
f. An animal is a cat if and only if it has four legs.
4. a. State the converse of the Isosceles Triangle Property Theorem.
b. Prove this converse.
We now consider the property to be an if and only if statement.
5. A new school is being built so that it will be equidistant from three small towns A, B, and C. If the distances between the towns are as shown in the diagram, determine an approximate location for the new school.
Thinking/Inquiry/
Problem Solving Application Communication Knowledge/
Understanding
Exercise 2.3
A
B C
3 km
4 km 6 km
6. Two sheds, S1and S2, are located in a circular com-pound as shown. Two gates are to be built so that each gate is equidistant from the two sheds. Where should the gates be located? Provide justification for your answer.
Part B
7. A river crosses two roads as shown. Determine the approximate location of a pumping station if the pumping station is equidistant from the two roads.
8. Find the location of a point that is equidistant from the two intersecting lines and is also equidistant from the two given points.
Provide a justification for your answer.
9. Suppose that you are given four points, A, B, C, and D. Explain why it is not likely that a circle would pass through the four points.
10. Y and Z are two given points on the circumference of a circle. Find a point X also on the circumference such that ∆XYZ is isosceles.
11. Prove that the right bisectors of the sides of a triangle pass through a common point.
12. Draw a series of five circles that pass through two given points.
a. Where do the centres of these circles lie?
b. What can you say about the centres of all circles that pass through the two given points? Why?
Thinking/Inquiry/
Problem Solving Thinking/Inquiry/
Problem Solving
S1
S2
River
A B
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Part C
13. Suppose we are given ∆ABC as shown. Show how to draw a line parallel to BC that meets AB at D and AC at E so that DE DB EC.
14. Suppose we are given ∆ABC as shown. Find a point D in side AB that is equidistant from A and the midpoint of BC.
15. Prove or disprove the following statement:
The angles in a triangle are in arithmetic sequence if and only if one of the angles equals 60º.
16. Prove that one of the roots of x3 ax2 bx c 0 is the negative of another if and only if c ab.
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C B
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