Squares square A square is a regular parallelogram. All angles are congruent All sides are congruent
Square Corollary A quadrilateral is a square if and only if it is a rhombus and a rectangle.
8.4 Properties of Rhombuses, Rectangles, and Squares Objectives: 1.To discover and use properties of rhombuses, rectangles, and squares 2.To find the area of rhombuses, rectangles, and squares
Example 2 Venn Diagram Below is a concept map showing the relationships between some members of the parallelogram family. This type of concept map is known as a Venn Diagram. Fill in the missing names.
Example 2 Venn Diagram Below is a concept map showing the relationships between some members of the parallelogram family. This type of concept map is known as a Venn Diagram.
Example 3 For any rhombus QRST, decide whether the statement is always or sometimes true. Draw a sketch and explain your reasoning. 1. Q S 2. Q R
Example 4 For any rectangle ABCD, decide whether the statement is always or sometimes true. Draw a sketch and explain your reasoning. 1. AB CD 2. AB BC
Example 5 Classify the special quadrilateral. Explain your reasoning.
Investigation 1 We know that the diagonals of parallelograms bisect each other. The diagonal of rectangles and rhombuses have a few other properties we will discover using GSP.
Diagonal Theorem 1 A parallelogram is a rectangle if and only if its diagonals are congruent.
Example 6 The previous theorem is a biconditional. Write the two conditional statements that must be proved separately to prove the entire theorem.
Example 7 You’ve just had a new door installed, but it doesn’t seem to fit into the door jamb properly. What could you do to determine if your new door is rectangular?
Diagonal Theorem 2 A parallelogram is a rhombus if and only if its diagonals are perpendicular.
Diagonal Theorem 3 A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles.
Example 8 Prove that if a parallelogram has perpendicular diagonals, then it is a rhombus. Given: ABCD is a parallelogram; AC BD Prove: ABCD is a rhombus
Example 9: SAT In the figure, a small square is inside a larger square. What is the area, in terms of x, of the shaded region?
Example 10 In the diagram below, MRVU SPTV. Let the area of MRVU equal A. Show that A = bh.
Rhombus Area Since a rhombus is a parallelogram, we could find its area by multiplying the base and the height.
Rhombus Area However, you’re not always given the base and height, so let’s look at the two diagonals. Notice that d 1 divides the rhombus into 2 congruent triangles. Ah, there’s a couple of triangles in there.
Rhombus Area So find the area of one triangle, and then double the result. Ah, there’s a couple of triangles in there.