Exercise 2 Page 407

Let's begin by drawing the diagram we made in the mentioned exercise.

This diagram can be read in two different directions.

1. Downward: a quadrilateral is sometimes a parallelogram; a parallelogram is sometimes a rhombus/rectangle; a rhombus/rectangle is sometimes a square.
2. Upward: a square is always a rhombus/rectangle; a rhombus/rectangle is always a parallelogram; a parallelogram is always a quadrilateral.

With that said, let's check and solve the statements given in exercises 65-70 by using the diagram above.

Exercise 65

A square is a rhombus. To relate a square to a rhombus, we need to read the diagram in an upward direction. Thus, the missing word is always. A square isalways a rhombus.

Exercise 66

A rectangle is a square. To relate a rectangle to a square, we need to read the diagram in a downward direction. Thus, the missing word is sometimes. A rectangle issometimes a square.

Exercise 67

A rectangle has congruent diagonals. Since a rectangle is always a parallelogram, we can apply the Rectangle Diagonals Theorem and conclude that its diagonals are congruent. Thus, the missing word is always. A rectangle always has congruent diagonals.

Exercise 68

The diagonals of a square bisect its angles. From the diagram, we know that a square is always a rhombus, so by the Rhombus Opposite Angles Theorem, we conclude that each diagonal bisects a pair of opposite angles. Thus, the missing word is always. The diagonals of a squarealways bisect its angles.

Exercise 69

A rhombus has four congruent angles. Recall that a square always has four congruent angles. Since a rhombus sometimes is a square, then we conclude that the missing word is sometimes. A rhombus sometimes has four congruent angles.

Exercise 70

A rectangle has perpendicular diagonals. As before, notice that a square always has perpendicular diagonals. Since a rectangle sometimes is a square, then we conclude that the missing word is sometimes. A rectangle sometimes has perpendicular diagonals.