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Warm-up

Warm-up. Pg 520 #39, 40 Pg 529 #43-45. Properties of Rhombuses, Rectangles, and Squares. 8.4. Rhombus. A rhombus is a parallelogram with four congruent sides. A quadrilateral is a rhombus if and only if it has four congruent sides. Rectangle.

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Warm-up

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  1. Warm-up • Pg 520 #39, 40 • Pg 529 #43-45

  2. Properties of Rhombuses, Rectangles, and Squares 8.4

  3. Rhombus • A rhombus is a parallelogram with four congruent sides. • A quadrilateral is a rhombus if and only if it has four congruent sides

  4. Rectangle • A rectangle is a parallelogram with four right angles. • A quadrilateral is a rectangle if and only if it has four right angles

  5. Square • A square is a parallelogram with four congruent sides and four right angles. • A quadrilateral is a square if and only if it is a rhombus and a rectangle

  6. Q Q a. S S a. By definition, a rhombus is a parallelogram with four congruent sides. By Theorem 8.4, opposite angles of a parallelogram are congruent. So, .The statement is always true. EXAMPLE 1 Use properties of special quadrilaterals For any rhombus QRST, decide whether the statement is always or sometimes true. Draw a sketch and explain your reasoning. SOLUTION

  7. Q b. R • If rhombus QRSTis a square, then all four angles are congruent right angles. So, if QRSTis a square. Because not all rhombuses are also squares, the statement is sometimes true. Q R EXAMPLE 1 Use properties of special quadrilaterals For any rhombus QRST, decide whether the statement is always or sometimes true. Draw a sketch and explain your reasoning. SOLUTION

  8. Classify the special quadrilateral. Explain your reasoning. EXAMPLE 2 Classify special quadrilaterals SOLUTION The quadrilateral has four congruent sides. One of the angles is not a right angle, so the rhombus is not also a square. By the Rhombus Corollary, the quadrilateral is a rhombus.

  9. 1. For any rectangle EFGH, is it always or sometimes true that Explain your reasoning. FG GH ? ANSWER Sometimes; this is only true if EFGH is a square. for Examples 1 and 2 GUIDED PRACTICE

  10. square ANSWER for Examples 1 and 2 GUIDED PRACTICE 2. A quadrilateral has four congruent sides and four congruent angles. Sketch the quadrilateral and classify it.

  11. Properties of Diagonals • A parallelogram is a rhombus if and only if its diagonals are perpendicular. • A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles. • A parallelogram is a rectangle if and only if its diagonals are congruent.

  12. EXAMPLE 3 List properties of special parallelograms Sketch rectangle ABCD. List everything that you know about it. SOLUTION • The figure has four right angles. • The figure is a parallelogram. Because ABCDis a parallelogram, it also has these properties: • Opposite sides are parallel and congruent. • Opposite angles are congruent. Consecutive angles are supplementary. • Diagonals bisect each other, and are congruent

  13. ANSWER Q P 1.PQRS is a parallelogram, rectangle and a rhombus.2. Opposite pairs of sides areparallel and all four sides arecongruent.3. All four angles are right angles.4. Diagonals are congruent and bisecteach other. R S for Example 3 GUIDED PRACTICE 3. Sketch square PQRS. List everything you know about the square.

  14. Carpentry You are building a frame for a window. The window will be installed in the opening shown in the diagram. EXAMPLE 4 Solve a real-world problem a. The opening must be a rectangle. Given the measurements in the diagram, can you assume that it is? Explain. b. You measure the diagonals of the opening. The diagonals are 54.8 inches and 55.3 inches. What can you conclude about the shape of the opening?

  15. Class/Group Work

  16. Pg. 538 • #26,28 • #32,34,36,38,40,42,44,46,48 • #53 • #55

  17. Homework • Pg 537 • 3-15 multof 3 • 19-23 odd • 27,29 • 33-48 mult of 3

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