6-4 Properties of Special Parallelograms Lesson Presentation Lesson Quiz HOMEWORK: Pg 413 #s 24-32 Pg 423 #s 18-26 Holt Geometry

Objectives Prove and apply properties of squares. Use properties of squares to solve problems. Prove that a given quadrilateral is a square.

A square is a quadrilateral with four right angles and four congruent sides. In the exercises, you will show that a square is a parallelogram, a rectangle, and a rhombus. So a square has the properties of all three.

Rectangles, rhombuses, and squares are sometimes referred to as special parallelograms. Helpful Hint

Example 3: Verifying Properties of Squares Show that the diagonals of square EFGH are congruent perpendicular bisectors of each other.

Example 3 Continued Step 1 Show that EG and FH are congruent. Since EG = FH,

Example 3 Continued Step 2 Show that EG and FH are perpendicular. Since ,

Example 3 Continued Step 3 Show that EG and FH are bisect each other. Since EG and FH have the same midpoint, they bisect each other. The diagonals are congruent perpendicular bisectors of each other.

SV ^ TW SV = TW = 122 so, SV @ TW . slope of TW = –11 1 slope of SV = Check It Out! Example 3 The vertices of square STVW are S(–5, –4), T(0, 2), V(6, –3) , and W(1, –9) . Show that the diagonals of square STVW are congruent perpendicular bisectors of each other. SV = TW = 122 so, SV @ TW . 1 11 slope of SV = slope of TW = –11 SV ^ TW

Check It Out! Example 3 Continued Step 1 Show that SV and TW are congruent. Since SV = TW,

Check It Out! Example 3 Continued Step 2 Show that SV and TW are perpendicular. Since

Check It Out! Example 3 Continued Step 3 Show that SV and TW bisect each other. Since SV and TW have the same midpoint, they bisect each other. The diagonals are congruent perpendicular bisectors of each other.

Lesson Quiz: Part III 5. The vertices of square ABCD are A(1, 3), B(3, 2), C(4, 4), and D(2, 5). Show that its diagonals are congruent perpendicular bisectors of each other.

In order to apply Theorems 6-5-1 through 6-5-5, the quadrilateral must be a parallelogram. Caution To prove that a given quadrilateral is a square, it is sufficient to show that the figure is both a rectangle and a rhombus. You will explain why this is true in Exercise 43.

You can also prove that a given quadrilateral is a rectangle, rhombus, or square by using the definitions of the special quadrilaterals. Remember!

Check It Out! Example 3A Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. K(–5, –1), L(–2, 4), M(3, 1), N(0, –4)

Check It Out! Example 3A Continued Step 1 Graph KLMN.

Check It Out! Example 3A Continued Step 2 Find KM and LN to determine is KLMN is a rectangle. Since , KMLN is a rectangle.

Check It Out! Example 3A Continued Step 3 Determine if KLMN is a rhombus. Since the product of the slopes is –1, the two lines are perpendicular. KLMN is a rhombus.

Check It Out! Example 3A Continued Step 4 Determine if PQRS is a square. Since PQRS is a rectangle and a rhombus, it has four right angles and four congruent sides. So PQRS is a square by definition.

Check It Out! Example 3B Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. P(–4, 6) , Q(2, 5) , R(3, –1) , S(–3, 0)