Download 5.3 Parallelograms and Rhombuses

5.3 Parallelograms and Rhombuses
parallelogram ______________________________________________________________________
1
In Chapter 3, we used the idea of symmetry to show that the
diagonal of a parallelogram divided the figure into two congruent
triangles. How can we prove that using what we know about
parallel lines?
2
3
4
This leads us to the next theorem…
Theorem 5.14
A diagonal of a parallelogram forms two congruent triangles.
B
B
C
A
C
A
D
If ABCD is a parallelogram, then
D
+ABD ≅ +CDB and +ABC ≅ +CDA
…and from this theorem, we can state the following corollaries.
Ú Corollary 5.15
(This gives us TWO sets of conditions that are true IF THE QUADRILATERAL IS A PARALLELOGRAM)
In a parallelogram, both pairs of opposite sides and opposite angles are congruent.
B
B
C
A
D
A
C
D
If ABCD is a parallelogram, then ∠A ≅ ∠C and ∠B ≅ ∠D
and if ABCD is a parallelogram, then AB ≅ DC and AD ≅ BC
Corollary 5.16
Parallel lines are everywhere equidistant.
a
A
C
If a & b , then AB=CD .
b
B
D
Proof: Given a & b , let AB and CD be two segments both perpendicular to line b. Because corresponding
angles are congruent, AB & CD and therefore ABCD is a parallelogram by the definition of parallelogram. Since
opposite sides of a parallelogram are congruent, AB=CD so the distance between the two lines is equal.
The following theorems are converses two conditionals contained in Corollary 5.15.
Corollary 5.17
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
B
C
Key Steps For Proof
+
+
Draw diagonal AC and prove
ABD – CDB by SSS.
pBCA –pDAC by CPCTC which means that BC & AD
because Alt. Int. angles are congruent.
A
Repeat this process after drawing diagonal BD to show
pCBD –pADB by CPCTC which will show AB & DC
thereby proving ABCD is a parallelogram by definition.
D
Corollary 5.18
If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
B
C
A
Key Steps For Proof
We will use the total number of degrees in a quadrilateral
(360E) and since pB –pD and pA –pC we have 2mpA +
2mpB = 360. Simplify this to mpA + mpB = 180 making
them supplementary. Then since same-side interior
angles are supplementary, one pair of sides is parallel.
Repeat to get the second pair of sides parallel which will
prove ABCD is a parallelogram by definition.
D
Theorem 5.19
If a quadrilateral has one pair of opposite sides both parallel and congruent, then the quadrilateral is a
parallelogram.
B
C
Proof is in the book on Page 260.
It uses congruent triangles again.
A
D
Theorem 5.20
Diagonals of a parallelogram bisect each other. (If it’s a parallelogram, then the diagonals bisect each other.)
B
C
Remember: If it’s a parallelogram, then both pairs of
opposite sides are congruent. Then you can prove
either + BMC – + DMA or + BMA – + DMC which will
M
give you BM ≅ MD and AM ≅ MC by CPCTC.
A
D
The converse is also true!
Theorem 5.21
If the diagonals of a quadrilateral bisect each other, then it’s a parallelogram.
What is a rhombus? ____________________________________________________________________
Theorem 5.22
Every rhombus is a parallelogram.
B
A
So everything that is true for parallelograms is true for rhombuses.
1. Diagonals form 2 congruent triangles
2. Both pairs of opposite angles (and sides) are congruent.
3. Diagonals bisect each other.
D
C
Theorem 5.23
The diagonals of a rhombus are perpendicular to each other.
B
A
So everything that is true for parallelograms is true for rhombuses.
1. Diagonals form 2 congruent triangles
2. Both pairs of opposite angles (and sides) are congruent.
3. Diagonals bisect each other.
D
C
And the converse…
Theorem 5.24
If the diagonals of a quadrilateral are perpendicular, then it is a rhombus.
The next theorem combines two conditionals together (we can also do that with Thm. 5.23 and Thm. 5.24)
Theorem 5.25
A parallelogram is a rhombus if and only if the diagonals bisect the opposite angles.
Examples from the book:
15)
2
1
3
55E
115E
17) The measures of consecutive angles in a parallelogram are in the ratio 2:3. Find the measures of
the angles of the parallelogram.
Hint for #46 - Read the Applied Problem and its solution.
the angle of incidence = the angle of reflection