Lesson 7.6
Parallelograms
pp. 291-295
Objectives:
1. To prove the SAS Congruence
Theorem for parallelograms.
2. To identify and prove the basic
properties of parallelograms.
Theorem 7.15
The opposite sides of a parallelogram
are congruent.
Theorem 7.15
The opposite sides of a parallelogram
are congruent.
Theorem 7.16
SAS Congruence for Parallelograms.
D
A
C
B
S
P
R
Q
Theorem 7.17
A quadrilateral is a parallelogram if
and only if the diagonals bisect one
another.
Theorem 7.18
Diagonals of a rectangle are
congruent.
Theorem 7.19
The sum of the measures of the four
angles of every convex quadrilateral
is 360°.
B
A
C
D
Theorem 7.20
Opposite angles of a parallelogram
are congruent.
Theorem 7.21
Consecutive angles of a
parallelogram are supplementary.
2
1
Theorem 7.22
If the opposite sides of a quadrilateral
are congruent, then the quadrilateral
is a parallelogram.
Theorem 7.23
A quadrilateral with one pair of
parallel sides that are congruent is a
parallelogram.
Practice: Determine whether the
given figure must be a parallelogram.
Be ready to explain your answer.
1. Yes
2. No
Practice: Determine whether the
given figure must be a parallelogram.
Be ready to explain your answer.
1. Yes D
2. No C
112°
68°
A
B
Practice: Determine whether the
given figure must be a parallelogram.
Be ready to explain your answer.
1. Yes
2. No
Practice: Determine whether the
given figure must be a parallelogram.
Be ready to explain your answer.
1. Yes
2. No
Practice: Determine whether the
given figure must be a parallelogram.
Be ready to explain your answer.
1. Yes
2. No
100°
80°
80°
100°
Practice: Determine whether the
given figure must be a parallelogram.
Be ready to explain your answer.
1. Yes
2. No
120°
60°
Homework
pp. 293-295
►A. Exercises
Using parallelogram ABCD, find the
measures of the indicated angles.
1. C
B
C
55º
48º
A
D
►A. Exercises
Using parallelogram ABCD, find the
measures of the indicated angles.
2. ADC
B
C
55º
48º
A
D
►A. Exercises
Using parallelogram ABCD, find the
measures of the indicated angles.
3. ABD
B
C
55º
48º
A
D
►A. Exercises
Using parallelogram ABCD, find the
measures of the indicated angles.
4. The four angles of the parallelogram
combined
B
C
55º
48º
A
D
►A. Exercises
Using parallelogram ABCD, find the
measures of the indicated angles.
5. An exterior angle of the parallelogram
at angle C
B
C
55º
48º
A
D
►A. Exercises
Disprove the following statements. Remember
that you disprove a statement by proving that it
is false. This can be done with an illustration or
a counterexample.
6. Adjacent angles form a linear pair.
►A. Exercises
Disprove the following statements. Remember
that you disprove a statement by proving that it
is false. This can be done with an illustration or
a counterexample.
7. Alternate interior angles are
congruent.
►A. Exercises
Disprove the following statements. Remember
that you disprove a statement by proving that it
is false. This can be done with an illustration or
a counterexample.
8. Every pair of supplementary angles
form a linear pair.
►A. Exercises
Disprove the following statements. Remember
that you disprove a statement by proving that it
is false. This can be done with an illustration or
a counterexample.
9. The acute angles of a triangle are
complementary.
►A. Exercises
Disprove the following statements. Remember
that you disprove a statement by proving that it
is false. This can be done with an illustration or
a counterexample.
10. If two triangles have a pair of
congruent angles, then the other
pairs of angles are congruent.
►B. Exercises
12. Given: ABCD with diagonals AC and BD
bisecting each other at E
Prove: ABCD is a parallelogram
B
C
E
A
D
►B. Exercises
12. Given: ABCD with diagonals AC and BD
bisecting each other at E
Prove: ABCD is a parallelogram
B
C
E
A
D
►B. Exercises
14. Given: Convex quadrilateral ABCD
Prove: mABC + mBCD + mCDA +
mDAB = 360º C
D
B
A
►B. Exercises
16. Given: ABCD is a parallelogram
Prove: A & B are supplementary
C
B
3
1
D
2
A
■ Cumulative Review
Suppose two segments must be proved
congruent. What reason could you use that
involves the concept named?
25. distances
■ Cumulative Review
Suppose two segments must be proved
congruent. What reason could you use that
involves the concept named?
26. bisectors
■ Cumulative Review
Suppose two segments must be proved
congruent. What reason could you use that
involves the concept named?
27. one triangle
■ Cumulative Review
Suppose two segments must be proved
congruent. What reason could you use that
involves the concept named?
28. two triangles
■ Cumulative Review
Suppose two segments must be proved
congruent. What reason could you use that
involves the concept named?
29. a circle
■ Cumulative Review
Suppose two segments must be proved
congruent. What reason could you use that
involves the concept named?
30. a parallelogram
Analytic Geometry
Midpoints
Midpoint Formula
If M is the midpoint of AB
where A(x1, y1) and B(x2, y2),
 x 1 + x 2 y1 + y 2 

then M = 
,
2 
 2
Find the coordinate of the midpoint
between the points (3, -5) and (1, -6).
 3 + 1 - 5 + ( - 6) 

M=
,
2
 2

 - 11 

=  2,
2 

Find the coordinate of the midpoint
between the two points.
1. (4, 8) and (2, -3)
Find the coordinate of the midpoint
between the two points.
2. (3, 5) and (3, 9)
Find the coordinate of the midpoint
between the two points.
3. (-1, -4) and (6, -2)