Question

1. a rhombus is a square.
2. a square is a rectangle. always true.

(a rectangle is a square sometimes true)

1. a diagonal divides a square into two isosceles right triangles.
2. opposite angles in a parallelogram are congruent.

Explanation:

Problem 6:

Step1: Recall rhombus and square definitions

A rhombus has all sides equal, opposite sides parallel, and opposite angles equal. A square has all sides equal, all angles \(90^\circ\), opposite sides parallel.

Step2: Determine if all rhombuses are squares

A rhombus with non - right angles (e.g., angles \(60^\circ\) and \(120^\circ\)) is not a square. Only rhombuses with right angles are squares. So "A rhombus is a square" is sometimes true.

Problem 7:

Step1: Recall square and rectangle definitions

A rectangle has opposite sides equal, all angles \(90^\circ\), opposite sides parallel. A square has all sides equal, all angles \(90^\circ\), opposite sides parallel.

Step2: Determine if all squares are rectangles

Since a square satisfies all the properties of a rectangle (right angles, opposite sides equal and parallel), "A square is a rectangle" is always true. For "A rectangle is a square", a rectangle with length \(5\) and width \(3\) (not equal sides) is not a square. Only rectangles with equal sides (squares) are squares. So "A rectangle is a square" is sometimes true.

Problem 8:

Step1: Recall square and triangle properties

A square has all sides equal, all angles \(90^\circ\). A diagonal of a square divides it into two triangles.

Step2: Analyze the triangles formed

The two triangles formed by a diagonal of a square have two sides equal (sides of the square) and the included angle \(90^\circ\). So they are isosceles right triangles (two equal sides, right angle). Thus, "A diagonal divides a square into two isosceles right triangles" is always true.

Problem 9:

Answer:

s:

1. Sometimes true
2. "A square is a rectangle": Always true; "A rectangle is a square": Sometimes true
3. Always true
4. Always true