Download Chapter 7 - Frost Middle School

Chapter 7
4. a.
L
M
Applications of Congruent Triangles
Lesson 7-7
(pp. 419–425)
Mental Math
a. a and b; c and d; e and f
b. Answers vary. Sample: a and c
c. 3-fold rotation symmetry
Q
P
b. By the Given and the Perpendicular to Parallels
Theorem, LMPQ must have four right angles,
so it must be a rectangle. Opposite sides of a
rectangle are equal in length.
c. rectangle
Activity 1
1. no; the parallelogram does not lie on itself when
folded over any of the line segments.
2. yes; ABCD has 2-fold rotation symmetry about
the point where the diagonals intersect.
3. Parallelograms only have rotation symmetry
about the point where the diagonals intersect.
Guided Example
a. 5
5. a. Corresponding Angles Postulate and Angle
Congruence Theorem
b. Parallel Lines Theorem (alternate interior
angles)
c. Properties of a Parallelogram Theorem, Part a
d. Reflexive Property of Congruence
e. SAS Congruence Theorem
f. CPCF Theorem
6. Consider parallelogram ABCD from Question 5. It
was shown that ADB CBD, and so by the
CPCF Theorem, ∠BAD ∠DCB. By the Parallel
Lines Theorem (alternate interior angles),
∠BCE ∠ABC. In Question 5, it was shown that
∠ADC ∠BCE, so by the Transitive Property of
Congruence, ∠ADC ∠ABC.
b. 8
c. 80
d. 100
Questions
1. a. 7
b. 5
c. 3
2. a. 5.5
b. 2x
3. a. ∠QSR
b. ∠USR
c. ∠QRU, ∠QSU
d. none
7. Answers vary. Sample:___
Consider
___parallelogram
ABCD with diagonals AC and BD intersecting
at E. From Part a of the Properties
___
___ of a
Parallelogram Theorem, AB CD. By the
Vertical Angles Theorem, m∠AEB = m∠CED.
So ∠AEB ∠CED ___
by the___
Angle Congruence
Theorem. Because AB CD, ∠BDC and ∠DBA
are congruent alternate interior angles (Parallel
Lines Theorem). So, AEB CED by the
AAS Congruence
Theorem.
the CPCF
___
___
___ By___
Theorem, AE CE and BE DE. Thus, by the
Segment Congruence Theorem, AE = CE and
BE = DE.
8. a. definition of parallelogram
b. Parallel Lines Theorem (alternate interior
angles)
c. definition of linear pair, Linear Pair Theorem
d. Supplements of congruent angles are congruent.
A128
Geometry
15. Conclusions
1. ABCD is a
parallelogram, and
M and N are the
___
midpoints
of AD
___
and BC
9. a. 64
b. 64
c. 90
d. 52
e. 26
___
___
2. AD BC
f. 26
10. a. LANO
____
b. 82
11. Distance between Parallel Lines Theorem
12. Conclusions
1.
2.
3.
4.
5.
6.
7.
8.
9.
Justifications
NTCR is a rectangle, Given
is the
and EA
perpendicular
___
bisector of RC.
___
___
RE EC
def. of midpoint
NTCR is a
Quadrilateral
parallelogram
Hierarchy Theorem
___ ___
NT RC
Prop. of a
Parallelogram
Theorem
___
___
EA ⊥ NT
⊥ to s Theorem
NAER and ATCE are def. of rectangle
rectangles.
___
___
___ ___
NA RE, AT EC def. of rectangle
___
___
NA AT
substitution
is the
def. of ⊥ bisector
EA
perpendicular
___
bisector of NT
13. a.–b.
parallelogram (R)
Justifications
Given
____
3. AM
MD,
___ ___
BN NC
4. AD = BC;
AM = MD;
BN = NC;
5. 2 · AM = 2 · BN
6. AM = BN
Prop. of a
Parallelogram
Theorem
def. of midpoint
Segment
Congruence
Theorem
substitution
Multiplication Prop.
of Equality
16. Answers vary. Sample:
17. Answers vary. Sample:
kite (rd)
18. true
rectangle (rpb, R)
rhombus (rd, R)
square (rd, rpb, R)
14. The three parallelograms are congruent;
therefore, AB = GH by the CPCF Theorem and
the Properties of a Parallelogram Theorem.
19. Answers vary. Sample: Draw the perpendicular
bisector for each side of the triangle. The center
of the circle in which the triangle is inscribed is
the intersection of the perpendicular bisectors.
20. always
21. C, D
22. a square
A129
Geometry