Section 8.4 Notes

Special Types of Parallelograms

1. A rhombus is a parallelogram with four congruent sides. 2 1. A rhombus is a parallelogram with four congruent sides. 2. A rectangle is a parallelogram with four right angles. 3. A square is a parallelogram with four congruent sides and four right angles.

Rhombus Corollary A quadrilateral is a rhombus if and only if it has four congruent sides. Rewrite the biconditional as two conditionals, then write the given and prove using the given diagram.

If a quadrilateral is a rhombus, then it has four congruent sides If a quadrilateral is a rhombus, then it has four congruent sides. Given: quad. ABCD is a rhombus If a quadrilateral has four congruent sides, then it is a rhombus. Prove: quad. ABCD is a rhombus

Rectangle Corollary A quadrilateral is a rectangle if and only if four right angles. Rewrite the biconditional as two conditionals, then write the given and prove using the given diagram.

If a quadrilateral is a rectangle, then it has four right angles If a quadrilateral is a rectangle, then it has four right angles. Given: quad. ABCD is a rectangle If a quadrilateral has four right angles, then it is a rectangle. Prove: quad. ABCD is a rectangle

Square Corollary A quadrilateral is a square if and only if it is a rhombus and a rectangle. Rewrite the biconditional as two conditionals, then write the given and prove using the given diagram.

If a quadrilateral is a square, then it is a rhombus and a rectangle If a quadrilateral is a square, then it is a rhombus and a rectangle. Given: quad. ABCD is a square Prove: quad. ABCD is a rhombus and a rectangle If a quadrilateral is a rhombus and a rectangle, then it is a square. Given: quad. ABCD is a rhombus and a rectangle Prove: quad. ABCD is a square

The Venn diagram on the next slide illustrates some important relationships among parallelograms, rhombuses, rectangles, and squares. For example, you can see that a square is a rhombus because it is a parallelogram with four congruent sides. Because it has four right angles, a square is also a rectangle.

Example 1 For any rectangle ABCD, decide whether the statement is always or sometimes true. Draw a sketch and explain your reasoning. This is always true because a rectangle is also a parallelogram by definition. A B C D

This is sometimes true if rectangle ABCD is a square. not true A B C D true

Example 2 For any rhombus RSTV, decide whether the statement is always or sometimes true. Draw a sketch and explain your reasoning.

a. S  V This is always true since RSTV is also a parallelogram.

b. T  V This is sometimes true when RSTV is a square. not true R S T V true

Example 3 Classify the special quadrilateral. Explain your reasoning. a. This is a rhombus. It is a parallelogram with consecutive sides congruent. Which means all sides are congruent.

b. This is a rhombus since it is a quadrilateral with 4 congruent sides but the four angles are not right angles.

Special Parallelogram Theorems

A parallelogram is a rhombus if and only if its diagonals are perpendicular. Rewrite the biconditional as two conditionals, then write the given and prove using the given diagram.

If a parallelogram is a rhombus, then its diagonals are perpendicular If a parallelogram is a rhombus, then its diagonals are perpendicular. Given: parallelogram ABCD is a rhombus If a parallelogram has diagonals that are perpendicular, then it is a rhombus. Prove: parallelogram ABCD is a rhombus

A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles. Rewrite the biconditional as two conditionals, then write the given and prove using the given diagram.

If a parallelogram is a rhombus, then each diagonal bisects opposite angles. Given: parallelogram ABCD is a rhombus If each diagonal of a parallelogram bisects opposite angles, then it is a rhombus. Prove: parallelogram ABCD is a rhombus

A parallelogram is a rectangle if and only if its diagonals are congruent. Rewrite the biconditional as two conditionals, then write the given and prove using the given diagram.

If a parallelogram is a rectangle, then its diagonals are congruent If a parallelogram is a rectangle, then its diagonals are congruent. Given: parallelogram ABCD is a rectangle If the diagonals of a parallelogram are congruent, then it is a rectangle. Prove: parallelogram ABCD is a rectangle

Example 4 Sketch a rhombus. List everything you know about it. R S T V

1. All sides are congruent. 2 1. All sides are congruent. 2. Both pairs of opposite sides are parallel. 3. Both pairs of opposite angles are congruent. 4. Diagonals bisect each other. 5. Diagonals are perpendicular. 6. Each diagonal bisects opposite angles.

Example 5 Sketch a rectangle. List everything you know about it. A B C D

1. All angles are right angles. 2 1. All angles are right angles. 2. Both pairs of opposite sides are parallel. 3. Both pairs of opposite sides are congruent. 4. Diagonals bisect each other. 5. Diagonals are congruent.

Example 6 Sketch a square. List everything you know about it. R S T V

1. All angles are right angles. 2. All sides are congruent. 3 1. All angles are right angles. 2. All sides are congruent. 3. Both pairs of opposite sides are parallel. 4. Diagonals bisect each other. 5. Diagonals are congruent. 6. Diagonals are perpendicular. 7. Each diagonal bisects opposite angles.