Download 7-8 answers - Frost Middle School

Chapter 7
Applications of Congruent Triangles
Lesson 7-8
(pp. 426–430)
Mental Math
a. yes
2. (a) A quadrilateral has one pair of parallel and
congruent sides. (b) Both pairs of opposite
sides of a quadrilateral are congruent. (c.) The
diagonals of a quadrilateral bisect each
other. (d) Both pairs of opposite angles of a
quadrilateral are congruent.
3. Construct quadrilateral ABCD with both pairs
of opposite
___sides congruent. Construct the
diagonal AC.
b. no
c. yes
A
Activity 1
Step 5.
D
yes
Activity 2
Conclusions
___
___ ___
___
1. AD BC, AB DC
___
___
2. AC AC
3. ADC CBA
4. ∠DCA ∠BAC
___ ___
5. AB DC
6. ∠CAD ∠ACB
___ ___
7. AB BC
8. ABCD is a
parallelogram.
Step 6.
C
A
P
B
D
Step 7.
C
A
C
Justifications
Given
Refl. Prop. of SSS Theorem
CPCF Theorem
Alt. Int. ∠s Theorem
CPCF Theorem
Alt. Int. ∠s Theorem
Def. of parallelogram
4. a. yes
P
B
b. This quadrilateral fulfills the sufficient
condition c, the diagonals bisect each other.
D
5. a. no
Step 8.
6. a. yes
C
A
P
B
D
Step 9.
The shape of the quadrilateral ACBD may
change,
but if P remains the midpoint of
___
___
AB and CD, then the quadrilateral remains a
parallelogram.
b. This quadrilateral fulfills the sufficient
condition d, both pairs of opposite angles are
congruent.
7. a. no
b. Draw the diagonal connecting the vertices
that have not been marked with angles, and
then there are two similar triangles (by SSA).
Alternate Interior Angles Theorem shows it is
a parallelogram.
8. a. no
Questions
1. A parallelogram is a quadrilateral with two pairs
of parallel sides.
A130
Geometry
9. a. Given:
Quadrilateral
___ ___
___
___ QUAD,
QU AD, QU AD.
Prove: QUAD is a parallelogram.
b. Conclusions
___
___
1. QU AD
2. ∠QUD ∠ADU
___
3.
4.
5.
6.
7.
___
QU AD
___
___
UD DU
QUD ADU
∠UDQ ∠DUA
___ ___
QD AU
8. QUAD is a
parallelogram.
13. Yes it is possible. Sample:
A
Justifications
Given
Lines Theorem
(alt. int. angles)
Given
Refl. Prop. of SAS Theorem
CPCF Theorem
Alternate Interior
Angles Theorem
def. of parallelogram
___
___
10. a. Given: Quadrilateral
___ABCD,
___AC BD = E, E
is the midpoint of AC and BD. Prove: ABCD is
a parallelogram.
b. Conclusions
1.
Quadrilateral
___
ABCD,
AC ___
BD = E, E is___
the
midpoint
___ of AC
and BD.
___
___ ___
AE EC, DE 2. ___
EB
3. ∠DEC ∠AEB
4. DEC AEB
5. ∠DCA ∠BAC
___ ___
6. AB DC
7. ∠DEA ∠BEC
8. DEA BEC
9. ∠ADB ∠CBD
___ ___
10. AD BC
11. ABCD is a
parallelogram.
Justifications
Given
def. of midpoint
Vertical ∠s Theorem
SAS Theorem
CPCF Theorem
Alt. Int. ∠s Theorem
Vertical ∠s Theorem
SAS Theorem
CPCF Theorem
Alt. Int. ∠s Theorem
def. of parallelogram
12. Construct the quadrilateral joining the edges
of the keyboard to the feet of the stand and
connecting the feet along the floor. By Part c
of the Sufficient Conditions for a Parallelogram
Theorem, the quadrilateral is a parallelogram, so
the keyboard will be parallel to the floor.
Geometry
150˚
C
150˚
30˚
D
B
14. a. One angle must be a right angle.
b. The diagonals must be congruent.
15. Yes it is possible.
96˚
84˚
84˚
96˚
16. a. ABCD is a parallelogram.
___
___
___
___
11
b. slope BC = slope AD = – __
, so AD BC. slope
___
___7 ___
___
AB = slope DC = 0, so AB DC. By definition,
ABCD is a parallelogram.
17. AD = 4 cm, DC = 1 cm, m∠C = 72,
m∠B = m∠D = 108
18. a.
11. rectangle, parallelogram, kite
A131
30˚
b.
19. Answers vary. Sample:
20. Answers vary. Sample:
21. Not true. A quadrilateral with these properties
that is not a parallelogram can be constructed.
Sample:
A
D
29.4˚
4.0 cm
4.0 cm
B
C
29.4˚