To determine which biconditional statements are true, let’s analyze each one individually:
1. "The number 117 is divisible by 9 if and only if the sum of the digits in 117 are evenly divisible by 9."
- The sum of the digits of 117 is [tex]\(1 + 1 + 7 = 9\)[/tex], which is divisible by 9. Thus, 117 is divisible by 9. This statement is true.
2. "A pentagon has five sides if and only if each angle measure is equal."
- A pentagon, by definition, has five sides. However, the angles in a pentagon are not necessarily equal; only a regular pentagon (an equilateral and equiangular pentagon) has all equal angles. Therefore, this statement is false.
3. "[tex]$10 + 10 = 20$[/tex] if and only if [tex]$2 \times 10 = 20$[/tex]."
- Both expressions evaluate to 20. So, if [tex]\(10 + 10 = 20\)[/tex], then [tex]\(2 \times 10 = 20\)[/tex] and vice versa. This statement is true.
4. "A rhombus is a square if and only if its opposite angles are equal."
- A rhombus has equal sides, but it is a square only if all angles are right angles, not only if the opposite angles are equal. Therefore, this statement is false.
5. "A number is divisible by 3 if and only if it's divisible by 6."
- A number divisible by 6 is always divisible by 3, but a number divisible by 3 is not necessarily divisible by 6. Therefore, this statement is false.
6. "The number 7 is even if and only if it's divisible by 2."
- The number 7 is odd, not even. Therefore, this statement is false.
From the analysis, the correct biconditional statements are:
1. "The number 117 is divisible by 9 if and only if the sum of the digits in 117 are evenly divisible by 9."
3. "[tex]$10 + 10 = 20$[/tex] if and only if [tex]$2 \times 10 = 20$[/tex]."
Thus, the true biconditional statements are statements 1 and 3.
