Mth 97
Winter 2013
Sections 5.3 and 5.4
Section 5.3 – Parallelograms and Rhombuses
Parallelograms
Theorem 5.14 – A diagonal of a parallelogram forms two congruent triangles.
A
B
If
then
C
D
Corollary 5.15 – In a parallelogram, the opposite sides are congruent and the opposite angles are
congruent.
A
B
If
then
C
D
Corollary 5.16 – Parallel lines are everywhere equidistant.
If
m
then
n
m
A
B
D
C
n
If
, then
Theorem 5.17 – If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is
a parallelogram.
Is the converse of 5.15?
A
B
If
then
D
C
Theorem 5.18 – If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral
is a parallelogram.
A
B
If
then
D
C
1
Mth 97
Winter 2013
Sections 5.3 and 5.4
Theorem 5.19 – If a quadrilateral has two sides that are parallel and congruent, then it is a parallelogram.
A
B
If
then
C
D
Theorem 5.20 – The diagonals of a parallelogram bisect each other.
A
B
A
B
M
C
D
C
If
then
D
Theorem 5.21 – If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
This theorem is the _______________________ of 5.20. Write the biconditional statement.
Rhombuses
E
Theorem 5.22 – Every rhombus is a parallelogram.
Previous theorems about parallelograms tell us…
1.
H
M
F
The diagonals of a rhombus bisect each other.
2. The opposite angles of a rhombus are congruent.
G
3. The diagonals of a rhombus form two congruent triangles.
Theorem 5.23 – The diagonals of a rhombus are perpendicular to each other.
A
If
D
B
then
C
Theorem 5.24 – If the diagonals of a parallelogram are perpendicular to each other, then the
parallelogram is a rhombus. Since this is the converse of theorem 5.23 write the biconditional statement.
2
Mth 97
Winter 2013
Sections 5.3 and 5.4
Theorem 5.25 – A parallelogram is a rhombus if and only if the diagonals bisect the opposite angles.
F
Write both statements that this biconditional statement says are true.
E
G
1)
H
2)
Do ICA 10
Section 5.4 – Rectangles, Squares and Trapezoids
Rectangles
Theorem 5.26 – Every rectangle is a parallelogram.
Previous theorems about parallelograms tell us…
1. The opposite sides are parallel.
A
B
M
D
C
2. The opposite sides are congruent.
3. The diagonals bisect each other.
Theorem 5.27 – A parallelogram with one right angle is a rectangle.
E
F
If EFGH is a parallelogram and ‫ﮮ‬F is a right angle,
then EFGH is a rectangle.
Given:
Prove: EFGH is a rectangle
H
G
Subgoals: Prove ‫ﮮ‬G is a right angle by
Prove ‫ﮮ‬E is a right angle by
Prove ‫ﮮ‬H is a right angle by
Write the inverse of Theorem 5.27 and tell whether it is true or false.
Write the contrapositive of Theorem 5.27 and tell whether it is true or false.
Theorem 5.28 – The diagonals of a rectangle are congruent.
S
T
If
then
V
U
Theorem 5.29 – If a parallelogram has congruent diagonals, then it is a rectangle.
Is this the converse of Theorem 5.28?
3
Mth 97
Winter 2013
Sections 5.3 and 5.4
Squares
Earlier we defined a square to be a quadrilateral with all sides congruent and four right angles.
Other possible definitions might be…
1. A square is a rhombus with one right angle.
A
B
2. A square is a rectangle with two adjacent sides congruent.
D
C
3. A square is a rhombus with congruent diagonals.
4. A square is a rectangle with perpendicular diagonals.
Isosceles Trapezoids
An isosceles trapezoid has congruent legs.
E
H
F
G
Theorem 5.30 – The base angles of an isosceles triangle are congruent.
If EFGH is a trapezoid with EF GH and _________________,then E  F and _______________
See page 273 for proof.
Theorem 5.31 is the converse of Theorem 5.30. Write it below.
Write the inverse of Theorem 5.30 and tell whether it is true or false.
Write the contrapositive of Theorem 5.30 and tell whether it is true or false.
4