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Understanding Rhombuses, Rectangles, and Squares in Advanced Geometry

Explore the unique properties of parallel and special quadrilaterals: rhombuses, rectangles, and squares. A rhombus features four congruent sides, while a rectangle possesses four right angles, and a square displays both characteristics. Learn key corollaries that help identify these shapes and gain insights into critical theorems about their diagonals and angles. This guide provides essential definitions, statements about special parallelograms, and examples to assess understanding of these fundamental geometric concepts.

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Understanding Rhombuses, Rectangles, and Squares in Advanced Geometry

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  1. 6.4 Rhombuses, Rectangles,and Squares Advanced Geometry

  2. Properties of Parallelograms Write down 5 properties of parallelograms.

  3. RHOMBUS A rhombus is a parallelogram with four congruent sides.

  4. RECTANGLE A rectangle is a parallelogram with four right angles.

  5. SQUARE A square is a parallelogram with four congruent sides and four right angles.

  6. Parallelograms

  7. Describing a Special Parallelogram Decide whether the statement is always, sometimes, or never true. • A rhombus is a rectangle • A parallelogram is a rectangle • A rectangle is a square • A square is a rhombus

  8. Corollaries about Special Quadrilaterals • Rhombus Corollary – A quadrilateral is a rhombus if and only if it has four congruent sides. • Rectangle Corollary – A quadrilateral is a rectangle if and only if it has four right angles. • Square Corollary – A quadrilateral is a square if and only if it is a rhombus and a rectangle.

  9. THEOREM 6.11 A parallelogram is a rhombus if and only if its diagonals are perpendicular.

  10. THEOREM 6.12 A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles.

  11. THEOREM 6.13 A parallelogram is a rectangle if and only if its diagonals are congruent.

  12. Example In parallelogram ABCD, the diagonals meet at point E, and AE = BE = 6. Is ABCD a rectangle?

  13. Example QRST is a square. What else do you know about QRST?

  14. Example EFGH is a rectangle. K is the midpoint of FH. If EG = 8z – 16, what is EK? GK?

  15. Example ABCD is a rectangle and measure of angle B = 8x + 26. What is the value of x?

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