Download 6-4 Special Parallelograms

6-4 SPECIAL PARALLELOGRAMS
(p. 312-318)
Note: Copy the pages with some of the proofs for the students.
The proofs for many of the following theorems can be presented by students at the board.
The students can prove these theorems by using the “tick mark” method rather than
writing a formal, complete proof.
Before the student presentations, the class will quickly examine each theorem to make
sure the class understands the terminology. The teacher will use diagrams and tick marks
to explain their meanings, but he will not prove the theorems.
Theorem 6-9
Each diagonal of a rhombus bisects two angles of the rhombus.
To prove Theorem 6-9, you can use congruent triangles (as the book does on p. 312) or
you can use the Isosceles Triangle Theorem (p. 211) and a parallel lines theorem to get
congruent angles.
Since a rhombus is equilateral (all sides are congruent), each vertex of the rhombus is
equidistant from the endpoints of the diagonal across from the vertex.
Example: Sketch a rhombus and one of its diagonals. Using tick marks, show why the
two opposite vertices are equidistant from the endpoints of the diagonal.
In section 5-2, we learned Theorem 5-3 (The Converse of the Perpendicular Bisector
Theorem)
If a point is equidistant from the endpoints of a segment, then it is on the
perpendicular bisector of the segment.
Short form: Equidistant pt.  pt. on  bis.
This chapter 5 theorem can help us prove the next theorem.
Theorem 6-10
The diagonals of a rhombus are perpendicular.
Theorem 6-10 can also be proved by using congruent triangles and the theorem that says
if two angles are both congruent and supplementary, then they are right angles.
Example: Find the measures of the numbered angles in rhombus ABCD.
B
C
1
68
4
2
3
A
5
D
Do 1 on p. 313.
The next theorem can be easily proved by using congruent triangles.
Theorem 6-11
The diagonals of a rectangle are congruent.
Example: One diagonal of a rectangle has length 8x  2 and the other diagonal has a
length 5x  11. Find the length of each diagonal by setting up and solving an equation.
The next three theorems are the converses of the first three theorems that we learned in
this lesson. These converses become methods of proving that certain parallelograms are
special parallelograms.
Theorem 6-12
If one diagonal of a parallelogram bisects two angles of the parallelogram, then
the parallelogram is a rhombus.
Have students complete the reasons for the following proof.
A
3
B
D
4
2 1
C
Given: ABCD is a parallelogram, AC bisects BAD and BCD
Prove: ABCD is a rhombus
STATEMENTS
1. ABCD is a parallelogram
2. AC bisects BAD and BCD
3. 1  2, 3  4
REASONS
1.
2.
3.
AC  AC
ABC  ADC
AB  AD
AB  DC, AD  BC
4.
5.
6.
7.
8. AB  BC  CD  AD
9. ABCD is a rhombus
8.
9.
4.
5.
6.
7.
Theorem 6-13
If the diagonals of a parallelogram are perpendicular, then the parallelogram is a
rhombus.
You will prove Theorem 6-13 as a future homework or quiz problem.
Theorem 6-14
If the diagonals of a parallelogram are congruent, then the parallelogram is a
rectangle.
Have students complete the reasons for the following proof.
B
C
A
D
Given: ABCD is a parallelogram, AC  BD
Prove: ABCD is a rectangle
STATEMENTS
1. ABCD is a parallelogram
2. AB  DC
3. AC  BD
4. AD  AD
5. BAD  CDA
6. BAD  CDA
7. BAD is supp. to CDA
8. BAD and CDA are right angles
9. BAD  BCD, ABC  CDA
10. BCD is a right angle
11. ABC is a right angle
12. ABCD is a rectangle
REASONS
1.
2.
3.
4.
5.
6.
7.
8.
9.
10. Substitution Property
11.
12.
You can use Theorems 6-12 through 6-14 to help you classify quadrilaterals.
Example: The diagonals of quadrilateral ABCD are perpendicular. AB = 15 cm and
BC = 7 cm. Can ABCD be a parallelogram? Explain.
Do 3 on p. 314.
Do 4 on p. 315. In order for the four students to create a square playing area, what must
they do to the ropes (think of the ropes as diagonals)? The ropes must satisfy three
conditions. What are they?
Homework p. 315-318: 5,7,13,14,16,20,21,24,27,32,33,35,38a,46,49,51,60,64,65,68
20. Yes. Since the angles are bisected, it could be a rhombus, which is a parallelogram.
21. Yes. Since the diagonals are perpendicular and it has a right angle, it could be a
square, which is a parallelogram.
35. Diagonals are congruent, diagonals are perpendicular
46. Yes. Since it has four congruent sides, it has both pairs of opposite sides congruent,
making it a parallelogram. A parallelogram with four congruent sides is a rhombus. A
good definition is reversible, precise, and uses clearly understood words. This statement
has these qualities.
49. This parallelogram is a square. Each diagonal of a rhombus bisects two angles of the
rhombus. A square is a rhombus.
Solve 9x = 45, 3y – 6 = 90, and 6z = 45.
51. Rectangle, square, isosceles trapezoid, kite