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main0 1 int i = 1, n, sum = 0; tout << "Enter a positive integer: tin >> n; while (i <= n) { sum += i*i; i++; > tout << "The sum of the first ' << n << ' squares is ' cc sum << endl;

PROOF:

II ;

Reasons 1. Through an external point, a line can be drawn parallel to a given line. 2. A straight angle is an angle whose measure is 180 . 3. The whole equals the sum of its parts. 4. Alternate interior angles of parallel lines are congruent. 5. Substitution Postulate

The first run computes the sum of the first 4 squares: 1 + 4 + 9 + 16 = 30. The second run computes the sum of the first 6 squares: 1 + 4 + 9 + 16 + 25 + 36 = 9 1.

Given: ^ABC, /A > /C To Prove: AB > BC Plan: When the bisector of /B is drawn, the sides to be proved congruent become corresponding sides of congruent triangles.

When you want several statements to execute within a loop, you need to use braces { > to combine them into a compound statement. Example 3.2 illustrates the standard way to format a compound statement in a loop. The left brace ends the loop s header line. The right brace stands on a line by itself directly below the w of the while keyword. And the statements within the compound statement are all indented. Of course, the compiler doesn t care how the code is formatted. It would accept this format:

PROOF:

while (i <= n) { sum += i*i; i-t+; }

'Area and Circumference of a Circle 'Averaging of Air Pollution Data Area = Pi * Radius ^ 2 Circumference = 2 * Pi * Radius 'Loop to Calculate Cumulative Sum Avg = Sum / n 'Calculate an Average Value 'Area 'Circumference

But most C++ programmers find the displayed format easier to read. Some C programmers also like to put the left brace on a line by itself, directly below the w of the while keyword.

Statements 1. 2. 3. 4. 5. 6. Draw BD bisecting /B. /1 > /2 /A > /C BD > BD ^BDA > ^BDC AB > BC 1. 2. 3. 4. 5. 6.

do statement while (condition);

Reasons An angle may be bisected. To bisect is to divide into two congruent parts. Given Reflexive property SAA Corresponding parts of congruent triangles are congruent.

The only difference is that the do. . . while statement executes the statement first and then evaluates the condition. These two steps are repeated until the condition evalzsto zero (i.e, is false). A do. . . whi 1 e loop will always iterate at least once, regardless of the value of the condi t ion, because the statement executes before the condi t ion is evaluated the first time.

4. Two right triangles are congruent if the hypotenuse and a leg of one are congruent to the corresponding parts of the other.

EXAMPLE 3.3 The Factorial Function This program computes the factorial function: IZ! = (n) main0 -C int n, f = 1;

Given: Right ^ABC with right angle at C Right ^DEF with right angle at F AB > DE, BC > EF To Prove: ^ABC > ^DEF Plan: Move the two given triangles together so that BC coincides with EF, forming an isosceles triangle. The given triangles are proved congruent by using Theorem 1 and SAA.

tout << 'Enter a positive integer: '; tout << n << ' factorial is '; do { f *= n; n--; } while (n > 1); tout << f << endl; tin >> n;

(~1 - 1) . . . (3) (2) ( 1) .

PROOF:

3

Enter a pus-itive integer: 5 _ !Ji EacturLgl is I.20 Enter a pusitive integer; 8

Statements 1. BC > EF 2. Move triangles ABC and DEF together so that BC coincides with EF, and A and D are on opposite sides of BC. 3. /C and /F are right angles. 4. /ACD is a straight angle. 5. AD is a straight line segment. 6. AB > DE 7. /A > /D 8. ^ABC > ^DEF