Transcript Chapter 4 Geometry

Chapter 4 Geometry
Homework Answers
Sec 4.1
2)
3)
4)
5)
6)
7)
8)
9)
73
60
110
24
3 x 360 – 180 = 900
3 x 180 – 180 = 360
a = 69, b = 47, c = 116, d = 93, e = 86
m = 30, n = 50, p = 82, q = 28, r = 32, s = 78,
t = 118, u = 50
14-16) See description.
17) False. See diagram.
18) False. See diagram.
19) False. See diagram.
20) False. See diagram.
21) True.
•
Answers in degrees unless labeled otherwise.
Sec 4.2
1)
2)
3)
4)
5)
6)
7)
79
16) Parallel
54
17) Parallel
107.5
18) Neither
44; 35 cm
21) New: (6, -3), (2, -5), (3, 0). Triangles
are congruent.
76; 3.5 cm
72; 10 cm
a = 124, b = 56, c = 56, d = 38, e = 38, 22) New: (3, -3), (-3, -1), (-1, -5). Triangles
are congruent.
f = 76,
g = 66, h = 104, k = 76, n
= 86, p = 38
8) a = 36, b = 36, c = 72, d = 108, e = 36;
none
10) NCA
• Answers in degrees unless labeled
11) IEC
otherwise.
15) Perpendicular
Using Your Algebra Skills 4: Writing
Linear Equations
1-3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
See graphs.
y = -x + 2
y = (-6/13)x + (74/13)
y = x+1
y = -3x +5
y = (2/5)x – (8/5)
y = 80 + 4x
y = -3x + 26
y = (-1/4)x – 3
y = (6/5)x
y=x+1
y = (-2/9)x + (43/9)
Sec 4.3
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
Yes
14)
No. See diagram.
15)
No. See diagram.
16)
Yes
17)
a, b, c
19)
c, b, a
20)
b, a, c
21)
a, c, b
22)
a, b, c
23)
v, z, y, w, x
6 < length < 102
By the Triangle Inequality Conjecture, the sum of
11 cm and 25 cm should be greater than 48 cm. •
13) By the Triangle Sum Conjecture, the third angle
must measure 36° in the small triangle, but it
measures 32° in the large triangle. These are the
same angle, so they can’t have different measures.
135
72
72
See description.
a = 52, b = 38, c = 110, d = 35
a = 90, b = 68, c = 112, d = 112, e = 68, f = 56,
g = 124, h = 124
ABE
FNK
cannot be determined
Answers in degrees unless labeled otherwise.
Sec 4.4
1)
2)
3)
4)
5)
6)
SAS
not correspond.
SSS
12)Cannot be determined. Parts do
not correspond.
cannot be determined
13)SAO by SAS
SSS
14)AIN by SSS or SAS
SAS
15)RAY by SAS
SSS (and the converse of the
Isosceles Triangle Conjecture)
16)The midpoint of SD and PR is (0, 0).
Therefore, ΔDRO ≈ ΔSPO by SAS.
8) b = 55, but 55 + 130 > 180, which is
impossible by the Triangle Sum
20)a = 37, b = 143, c = 37, d = 58, e =
Conjecture.
37, f = 53, g = 48, h = 84, k = 96, m
= 26, p = 69, r = 111, s = 69
9) FLE by SSS
10)Cannot be determined. SSA is not a
congruence conjecture.
• Answers in degrees unless labeled
otherwise.
11)Cannot be determined. Parts do
Sec 4.5
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
ASA
ASA
SAA
Cannot be determined.
Cannot be determined.
Cannot be determined.
FED by SSS
WTA by ASA or SAA
SAT by SAS
PRN by ASA or SAS; SRE by ASA
Cannot be determined. Parts do
not correspond.
12) MRA by SAS
13) Cannot be determined. AAA does
not guarantee congruence.
14) WKL by ASA
15) Yes, three exterior triangles are
congruent by SAS.
Sec 4.6
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
Yes; AAS
Yes; ASA
Cannot be determined.
Yes; AAS
Yes; SSS
Yes; SAS
Yes; SSS
Cannot be determined.
Cannot be determined.
Yes. The triangles are congruent by SAS.
Yes. The triangles are congruent by SAS, and
the angles are congruent by CPCTC.
See description.
Cannot be determined.
KEI by ASA
UTE by SAS
18)
23)
a = 112, b = 68, c = 44, d = 44, e = 136, f = 68, g =
68, h = 56, k = 68, l = 56, m = 124
x = 3, y = 10
•
Answers to # 18 are in degrees.
Sec 4.7
1)
2)
3)
4)
5)
6)
8)
The angle bisector does not go to the midpoint
of the opposite side in any triangle, only in an
isosceles triangle.
1. Given, 2. Given, 3. Vertical Angles Conjecture,
4. Δ ESM ≈ Δ USO, 5. CPCTC
2. Given, 4. Definition of midpoint, 5. Vertical
Angles Conjecture, 6. Δ CIL ≈ Δ MIB by SAS,
7. CL ≈ MB
5. Given, 6. Δ ESN by SSS, 7. ‹ E by CPCTC
2. ‹ 2 is an angle bisector, 3. Given, 4. Same
segment, 5. ΔWNS ≈ ΔENS by SAA, 6. CPCTC,
7. Definition of isosceles triangle
1. Given, 2. Given, 4. ‹ 1 ≈ ‹ 2 by AIA Conjecture,
6. Δ ESN ≈ Δ ANS by ASA, 7. SA ≈ NE by CPCTC
– This proof shows that in a parallelogram,
opposite sides are congruent.
NE, because it is across from the smallest angle
in Δ NAE. It is shorter than AE, which is across
from the smallest angle in Δ LAE.
9)
10)
11)
12)
13)
•
The triangles are congruent by SSS, so the two
central angles cannot have different measures.
PRN by ASA, SRE by ASA
Cannot be determined. Parts do not correspond.
ACK by SSS
a = 72, b = 36, c = 144, d = 36, e = 144, f = 18, g =
162, h = 144, j = 36, k = 54, m = 126
Answers to # 13 are in degrees.
Sec 4.8
1)
2)
3)
4)
5)
6
= 52, j = 70, k = 70, l = 40, m =
110, n = 58
90°
45°
2. ‹ 1 ≈ ‹ 2, 5. SAS
4. CPCTC, 7. Definition of
perpendicular, 8. CD is an
• Answers to # 11 are in
altitude
degrees.
6) See diagram.
11)a = 128, b = 128, c = 52, d =
76, e = 104, f = 104, g = 76, h
Ch 4 Review
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
Their rigidity gives strength
The Triangle Sum Conjecture states that the sum of the measures of the angles in every triangle is 180°.
Possible answers: It applies to all triangles; many other conjectures rely on it.
The angle bisector of the vertex angle is also the median and the altitude.
The distance between A and B is along the segment connecting them. The distance from A to C to B
can’t be shorter than the distance from A to B. Therefore AC + CB > AB. Points A, B, and C form a
triangle. Therefore the sum of the lengths of any two sides is greater than the length of the third side.
SSS, SAS, ASA, or SAA
In some cases two different triangles can be constructed using the same two sides and non-included
angle.
Cannot be determined
ΔTOP ≅ ΔZAP by SAA
ΔMSE ≅ ΔOSU by SSS
Cannot be determined
ΔTRP ≅ ΔAPR by SAS
Cannot be determined
ΔCGH ≅ ΔNGI by SAS
ΔABE ≅ ΔDCE by SAA or ASA
ΔACN ≅ ΔRBO or OBR by SAS
ΔAMD ≅ ΔUMT by SAS, AD ≅ UT by CPCTC
Cannot be determined
Ch 4 Review cont.
18)
19)
20)
21)
22)
23)
24)
25)
Cannot be determined
ΔTRI ≅ ΔALS by SAA, TR ≅ AL by CPCTC
ΔSVE ≅ ΔNIK by SSS, EI ≅ KV by overlapping segments property
Cannot be determined
Cannot be Determined
ΔLAZ ≅ ΔIAR by ASA, ΔLRI ≅ ΔIZL by ASA, ΔLRD ≅ ΔIZD by ASA
ΔPTS ≅ ΔTPO by ASA or SAA
ΔANG is isosceles, so ∠𝐴 ≅ ∠𝐺. However, the sum of m∠𝐴 + m∠𝑁 + m∠𝐺 = 188°. The measures of the
three angles of a triangle must sum to 180°.
26) ΔROW ≅ ΔNOG by ASA, implying that OW ≅ OG. However, the two segments shown are not equal in
measure.
27) a = g < e = d = b = f < c. Thus, c is the longest segment and a and g are the shortest.
28) X = 20°