Title: 6.2 Properties of Parallelograms
16.2 Properties of Parallelograms
- Geometry
- Mrs. Spitz
- Spring 2005
2Objectives
- Use some properties of parallelograms.
- Use properties of parallelograms in real-lie
situations such as the drafting table shown in
example 6.
3Assignment
4In this lesson . . .
- And the rest of the chapter, you will study
special quadrilaterals. A parallelogram is a
quadrilateral with both pairs of opposite sides
parallel. - When you mark diagrams of quadrilaterals, use
matching arrowheads to indicate which sides are
parallel. For example, in the diagram to the
right, PQRS and QRSP. The symbol PQRS is
read parallelogram PQRS.
5Theorems about parallelograms
Q
R
- 6.2If a quadrilateral is a parallelogram, then
its opposite sides are congruent. - ?PQ?RS and SP?QR
P
S
6Theorems about parallelograms
Q
R
- 6.3If a quadrilateral is a parallelogram, then
its opposite angles are congruent. - ?P ? ?R and
- ?Q ? ?S
P
S
7Theorems about parallelograms
Q
R
- 6.4If a quadrilateral is a parallelogram, then
its consecutive angles are supplementary (add up
to 180). - m?P m?Q 180,
- m?Q m?R 180,
- m?R m?S 180,
- m?S m?P 180
P
S
8Theorems about parallelograms
Q
R
- 6.5If a quadrilateral is a parallelogram, then
its diagonals bisect each other. - QM ? SM and
- PM ? RM
P
S
9Ex. 1 Using properties of Parallelograms
5
G
F
- FGHJ is a parallelogram. Find the unknown
length. Explain your reasoning. - JH
- JK
3
K
H
J
b.
10Ex. 1 Using properties of Parallelograms
5
G
F
- FGHJ is a parallelogram. Find the unknown
length. Explain your reasoning. - JH
- JK
- SOLUTION
- a. JH FG Opposite sides of a are ?.
- JH 5 Substitute 5 for FG.
3
K
H
J
b.
11Ex. 1 Using properties of Parallelograms
5
G
F
- FGHJ is a parallelogram. Find the unknown
length. Explain your reasoning. - JH
- JK
- SOLUTION
- a. JH FG Opposite sides of a are ?.
- JH 5 Substitute 5 for FG.
3
K
H
J
b.
- JK GK Diagonals of a bisect each
other. - JK 3 Substitute 3 for GK
12Ex. 2 Using properties of parallelograms
R
Q
- PQRS is a parallelogram.
- Find the angle measure.
- m?R
- m?Q
70
P
S
13Ex. 2 Using properties of parallelograms
R
Q
- PQRS is a parallelogram.
- Find the angle measure.
- m?R
- m?Q
- a. m?R m?P Opposite angles of a are ?.
- m?R 70 Substitute 70 for m?P.
70
P
S
14Ex. 2 Using properties of parallelograms
R
Q
- PQRS is a parallelogram.
- Find the angle measure.
- m?R
- m?Q
- a. m?R m?P Opposite angles of a are ?.
- m?R 70 Substitute 70 for m?P.
- m?Q m?P 180 Consecutive ?s of a are
supplementary. - m?Q 70 180 Substitute 70 for m?P.
- m?Q 110 Subtract 70 from each side.
70
P
S
15Ex. 3 Using Algebra with Parallelograms
P
Q
- PQRS is a parallelogram. Find the value of x.
- m?S m?R 180
- 3x 120 180
- 3x 60
- x 20
3x
120
S
R
- Consecutive ?s of a ? are supplementary.
- Substitute 3x for m?S and 120 for m?R.
- Subtract 120 from each side.
- Divide each side by 3.
16Ex. 4 Proving Facts about Parallelograms
- Given ABCD and AEFG are parallelograms.
- Prove ?1 ? ?3.
- ABCD is a ?. AEFG is a ?.
- ?1 ? ?2, ?2 ? ?3
- ?1 ? ?3
- Given
17Ex. 4 Proving Facts about Parallelograms
- Given ABCD and AEFG are parallelograms.
- Prove ?1 ? ?3.
- ABCD is a ?. AEFG is a ?.
- ?1 ? ?2, ?2 ? ?3
- ?1 ? ?3
- Given
- Opposite ?s of a ? are ?
18Ex. 4 Proving Facts about Parallelograms
- Given ABCD and AEFG are parallelograms.
- Prove ?1 ? ?3.
- ABCD is a ?. AEFG is a ?.
- ?1 ? ?2, ?2 ? ?3
- ?1 ? ?3
- Given
- Opposite ?s of a ? are ?
- Transitive prop. of congruence.
19Ex. 5 Proving Theorem 6.2
- Given ABCD is a parallelogram.
- Prove AB ? CD, AD ? CB.
- ABCD is a ?.
- Draw BD.
- AB CD, AD CB.
- ?ABD ? ?CDB, ?ADB ? ? CBD
- DB ? DB
- ?ADB ? ?CBD
- AB ? CD, AD ? CB
- Given
20Ex. 5 Proving Theorem 6.2
- Given ABCD is a parallelogram.
- Prove AB ? CD, AD ? CB.
- ABCD is a ?.
- Draw BD.
- AB CD, AD CB.
- ?ABD ? ?CDB, ?ADB ? ? CBD
- DB ? DB
- ?ADB ? ?CBD
- AB ? CD, AD ? CB
- Given
- Through any two points, there exists exactly one
line.
21Ex. 5 Proving Theorem 6.2
- Given ABCD is a parallelogram.
- Prove AB ? CD, AD ? CB.
- ABCD is a ?.
- Draw BD.
- AB CD, AD CB.
- ?ABD ? ?CDB, ?ADB ? ? CBD
- DB ? DB
- ?ADB ? ?CBD
- AB ? CD, AD ? CB
- Given
- Through any two points, there exists exactly one
line. - Definition of a parallelogram
22Ex. 5 Proving Theorem 6.2
- Given ABCD is a parallelogram.
- Prove AB ? CD, AD ? CB.
- ABCD is a ?.
- Draw BD.
- AB CD, AD CB.
- ?ABD ? ?CDB, ?ADB ? ? CBD
- DB ? DB
- ?ADB ? ?CBD
- AB ? CD, AD ? CB
- Given
- Through any two points, there exists exactly one
line. - Definition of a parallelogram
- Alternate Interior ?s Thm.
23Ex. 5 Proving Theorem 6.2
- Given ABCD is a parallelogram.
- Prove AB ? CD, AD ? CB.
- ABCD is a ?.
- Draw BD.
- AB CD, AD CB.
- ?ABD ? ?CDB, ?ADB ? ? CBD
- DB ? DB
- ?ADB ? ?CBD
- AB ? CD, AD ? CB
- Given
- Through any two points, there exists exactly one
line. - Definition of a parallelogram
- Alternate Interior ?s Thm.
- Reflexive property of congruence
24Ex. 5 Proving Theorem 6.2
- Given ABCD is a parallelogram.
- Prove AB ? CD, AD ? CB.
- ABCD is a ?.
- Draw BD.
- AB CD, AD CB.
- ?ABD ? ?CDB, ?ADB ? ? CBD
- DB ? DB
- ?ADB ? ?CBD
- AB ? CD, AD ? CB
- Given
- Through any two points, there exists exactly one
line. - Definition of a parallelogram
- Alternate Interior ?s Thm.
- Reflexive property of congruence
- ASA Congruence Postulate
25Ex. 5 Proving Theorem 6.2
- Given ABCD is a parallelogram.
- Prove AB ? CD, AD ? CB.
- ABCD is a ?.
- Draw BD.
- AB CD, AD CB.
- ?ABD ? ?CDB, ?ADB ? ? CBD
- DB ? DB
- ?ADB ? ?CBD
- AB ? CD, AD ? CB
- Given
- Through any two points, there exists exactly one
line. - Definition of a parallelogram
- Alternate Interior ?s Thm.
- Reflexive property of congruence
- ASA Congruence Postulate
- CPCTC
26Ex. 6 Using parallelograms in real life
- FURNITURE DESIGN. A drafting table is made so
that the legs can be joined in different ways to
change the slope of the drawing surface. In the
arrangement below, the legs AC and BD do not
bisect each other. Is ABCD a parallelogram?
27Ex. 6 Using parallelograms in real life
- FURNITURE DESIGN. A drafting table is made so
that the legs can be joined in different ways to
change the slope of the drawing surface. In the
arrangement below, the legs AC and BD do not
bisect each other. Is ABCD a parallelogram? - ANSWER NO. If ABCD were a parallelogram, then
by Theorem 6.5, AC would bisect BD and BD would
bisect AC. They do not, so it cannot be a
parallelogram.