Holt Geometry

6-4 Properties of Special Parallelograms6-4 Properties of Special Parallelograms

Holt Geometry

Lesson PresentationLesson Presentation

Lesson QuizLesson QuizHOMEWORK:HOMEWORK:

Pg 413 #s 24-32Pg 413 #s 24-32Pg 423 #s 18-26Pg 423 #s 18-26

Holt Geometry

6-4 Properties of Special Parallelograms

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

Objectives

Holt Geometry

6-4 Properties of Special Parallelograms

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.

Holt Geometry

6-4 Properties of Special Parallelograms

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

Helpful Hint

Holt Geometry

6-4 Properties of Special ParallelogramsExample 3: Verifying Properties of Squares

Show that the diagonals of square EFGH are congruent perpendicular bisectors of each other.

Holt Geometry

6-4 Properties of Special ParallelogramsExample 3 Continued

Step 1 Show that EG and FH are congruent.

Since EG = FH,

Holt Geometry

6-4 Properties of Special ParallelogramsExample 3 Continued

Step 2 Show that EG and FH are perpendicular.

Since ,

Holt Geometry

6-4 Properties of Special Parallelograms

The diagonals are congruent perpendicular bisectors of each other.

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.

Holt Geometry

6-4 Properties of Special ParallelogramsCheck 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.

111slope of SV =

slope of TW = –11

SV TW

SV = TW = 122 so, SV TW .

Holt Geometry

6-4 Properties of Special Parallelograms

Step 1 Show that SV and TW are congruent.

Check It Out! Example 3 Continued

Since SV = TW,

Holt Geometry

6-4 Properties of Special Parallelograms

Step 2 Show that SV and TW are perpendicular.

Check It Out! Example 3 Continued

Since

Holt Geometry

6-4 Properties of Special Parallelograms

The diagonals are congruent perpendicular bisectors of each other.

Step 3 Show that SV and TW bisect each other.

Since SV and TW have the same midpoint, they bisect each other.

Check It Out! Example 3 Continued

Holt Geometry

6-4 Properties of Special ParallelogramsLesson 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.

Holt Geometry

6-4 Properties of Special Parallelograms

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.

Holt Geometry

6-4 Properties of Special Parallelograms

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

Remember!

Holt Geometry

6-4 Properties of Special ParallelogramsCheck 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)

Holt Geometry

6-4 Properties of Special Parallelograms

Step 1 Graph KLMN.

Check It Out! Example 3A Continued

Holt Geometry

6-4 Properties of Special ParallelogramsCheck It Out! Example 3A Continued

Step 2 Find KM and LN to determine is KLMN is a rectangle.

Since , KMLN is a rectangle.

Holt Geometry

6-4 Properties of Special Parallelograms

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

Holt Geometry

6-4 Properties of Special Parallelograms

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 3A Continued

Holt Geometry

6-4 Properties of Special ParallelogramsCheck 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)