Here are the True/False evaluations for each statement:

1. False – A line perpendicular to another does not necessarily bisect it. Perpendicularity only guarantees a right angle, not bisection.

2. True – An equilateral triangle is also equiangular, meaning all angles are equal to 60°.

3. False – The altitude of a triangle does not always pass through the midpoint of a side. This is true only for certain triangles like equilateral triangles.

4. True – In an isosceles triangle, the median to the base is also the altitude and perpendicular to the base.

5. False – The bisector of an angle of a triangle does not necessarily bisect the side opposite; it divides the opposite side proportionally to the other two sides, not bisects it.

6. False – The altitude does not necessarily intersect the midpoint of a side. This is specific to equilateral or isosceles triangles.

7. False – The bisectors of two angles of a triangle are not necessarily perpendicular to each other.

8. True – In an equilateral triangle, the altitude is a perpendicular bisector of the base.

9. True – In an equilateral triangle, all three angles, including the base angles, are congruent.

10. True – In an isosceles triangle, all three angles are acute, unless the triangle is specifically designed to be obtuse.

11. False – If the two diagonals of a quadrilateral are perpendicular, the quadrilateral is not necessarily a parallelogram; it could be a kite.

12. False – A parallelogram is not necessarily a rectangle. While a rectangle is a type of parallelogram, the reverse is not true unless all angles are 90°.

13. True – A square is a special type of rectangle, as it has all properties of a rectangle (four right angles and opposite sides parallel), with the additional property of all sides being equal.

14. True – An isosceles trapezoid has two congruent non-parallel sides.

15. True – The two diagonals of a rhombus bisect each other at right angles.

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Would you like further details or explanations on any of these statements?

Here are five related questions to extend your understanding:

1. What is the difference between a parallelogram and a rhombus?
2. Can a quadrilateral have diagonals that bisect each other but not be a parallelogram?
3. How does the definition of an isosceles triangle differ from that of an equilateral triangle?
4. Why do the bisectors of two angles of a triangle not necessarily meet perpendicularly?
5. In what ways does the altitude of a triangle vary depending on its type?

Tip: Always check geometric definitions carefully—terms like "bisect," "perpendicular," and "congruent" have very specific meanings in geometry!