Question

Which biconditional statement is not true? a) A triangle is equilateral if and only if it is equiangular. b) A triangle is a right triangle if and only if the lengths of its sides are related by the equation a^2 + b^2 = c^2. c) Two lines are perpendicular if and only if they intersect to form right angles. d) A quadrilateral is a rectangle if and only if it has at least two right angles.

Answer 1

The biconditional statement that is not true is option b) A triangle is a right triangle if and only if the lengths of its sides are related by the equation a^2 + b^2 = c^2.

Step-by-step explanation:

Option b) suggests that a triangle is a right triangle if and only if its side lengths are related by the Pythagorean theorem, a^2 + b^2 = c^2. However, this statement is not accurate. While the Pythagorean theorem holds true for right triangles, not all triangles that satisfy the theorem are necessarily right triangles. The Pythagorean theorem is a condition for right triangles, but it is not the only condition.

For instance, consider a triangle with side lengths of 3, 4, and 5. When using the Pythagorean theorem, 3^2 + 4^2 = 9 + 16 = 25, which satisfies a^2 + b^2 = c^2. This satisfies the condition proposed in option b). However, this triangle is indeed a right triangle, fulfilling the Pythagorean condition, but not all triangles satisfying this equation are right triangles. There are other triangles (like equilateral or scalene triangles) that can also satisfy this equation but are not right triangles.

Therefore, the biconditional statement in option b) is not true because while the Pythagorean theorem is a condition for right triangles, it does not exclusively define all right triangles, as other types of triangles may also satisfy this equation without being right triangles.

Answer 2

The biconditional statement that is not true is: b) A triangle is a right triangle if and only if the lengths of its sides are related by the equation
`\(a^2 + b^2 = c^2\)`.

This statement is a misrepresentation of the Pythagorean Theorem. The correct formulation of the Pythagorean Theorem is
`\(c^2 = a^2 + b^2\)`, where
`\(c\)` is the length of the hypotenuse and
`\(a\)` and
`\(b\)` are the lengths of the other two sides in a right-angled triangle.

So, the correct biconditional statement related to the Pythagorean Theorem is: A triangle is a right triangle if and only if the lengths of its sides are related by the equation
`\(c^2 = a^2 + b^2\)`.