1. Lesson 5-4: Special Quadrilaterals (page 184)
Essential Question
How can the properties of quadrilaterals be used to solve real life problems?

2. Both pairs of opposite ∠’s are ≅
RECTANGLE: a quadrilateral with four right angles.
Every rectangle is a _____;
reason:____________________________________________.
Both pairs of opposite ∠’s are ≅

3. The diagonals of a rectangle are congruent .
Theorem 5-12
The diagonals of a rectangle are congruent . A B D C

4. Theorem 5-16
If an angle of a parallelogram is a right angle, then the parallelogram is a rectangle. W X Z Y
Proof: Show that all 4 angles are right angles, then the parallelogram is a rectangle by the Definition of a Rectangle .

5. Both pairs of opposite sides are ≅
RHOMBUS: a quadrilateral with four congruent sides.
Every rhombus is a _____;
reason:__________________________________________.
Both pairs of opposite sides are ≅

6. The diagonals of a rhombus are perpendicular .
Theorem 5-13
The diagonals of a rhombus are perpendicular . A X D B C

7. Each diagonal of a rhombus bisects two angles of the rhombus.
Theorem 5-14
Each diagonal of a rhombus bisects two angles of the rhombus. A X D B C
Proof: Show ∆AXD ≅ ∆AXB ≅ ∆CXD ≅ ∆CXB,
by the SSS Postulate , then use CPCTC to prove …
∠DAX ≅ ∠BAX , ∠DCX ≅ ∠BCX ,
∠ABX ≅ ∠CBX, and ∠CDX ≅ ∠ADX.

8. Theorem 5-17
If two consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus. A X D B C
Proof: Show that all 4 sides are congruent , then the parallelogram is a rhombus by the Definition of a Rhombus .

9. I’m a rectangle too! I’m a rhombus too!
SQUARE: a quadrilateral with four right angles
and four congruent sides.
I’m a
rectangle
too!
I’m a
rhombus
too!
Every square is a _______;
reason:____________________________________,
__________________________________________
Both pairs of opposite ∠’s are ≅
or both pairs of opposite sides are ≅ .

10. Summary of Special Parallelograms
RECTANGLE: has four right angles.
has all the properties of a parallelogram.
has diagonals that are congruent.
RHOMBUS: has four congruent sides.
has diagonals that are perpendicular.
has diagonals bisect its angles.
SQUARE: has four right angles and four congruent sides.
has diagonals that are both congruent and perpendicular.
has all the properties of a rectangle and rhombus.

11. Theorem 5-15 circumcenter Perpendicular - Bisector
The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices. A
Given: Right ∆ ABC
X is the midpoint of AB
Prove: XA = XB = XC X ● C B
circumcenter
The name given to point X is the ____________________, which is the
intersection of the ________________________________ of each side.
Perpendicular - Bisector

12. This is a rhombus, because …
Example #1: State which kind of special quadrilateral this diagram represents.
This is a rhombus, because …
all 4 sides are congruent … definition of a rhombus.

13. This is a rectangle, because …
Example #2: State which kind of special quadrilateral this diagram represents. ➤ ➤
This is a rectangle, because …
one pair of opposite sides is both parallel and congruent,
so the quadrilateral is a parallelogram.
And a parallelogram with one right angle is a rectangle.

14. This is a rectangle, because …
Example #3: State which kind of special quadrilateral this diagram represents.
This is a rectangle, because …
all 4 angles are right angles … definition of a rectangle.

15. This is a rhombus, because …
Example #4: State which kind of special quadrilateral this diagram represents.
This is a rhombus, because …
the diagonals bisect each so the quadrilateral is a parallelogram. And a parallelogram with consecutive sides congruent is a rhombus.

16. Assignment
Written Exercises on pages 187 & 188
RECOMMENDED: 1 to 10 (copy & complete the chart)
29, 36 (a), 37
REQUIRED: 13 to 27 odd numbers, 38
You will need graph paper for this assignment.
How can the properties of quadrilaterals be used to solve real life problems?