Download 4.1 Triangles Sum Conjectures

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4.1 Triangles Sum Conjectures
Auxillary line: an extra line or segment that helps you with your proof.
Page 202 Paragraph proof explaining why the Triangle Sum Conjecture is true.
Conjecture: The sum of the measures of the angles in every triangle is 180°.
HJJG HJJG
ΔABC with auxiliary line EC & AB and
Given:
angles labeled as shown.
Show:
m∠ 2 + m∠ 4 + m∠5 = 180°
m∠1 + m∠ 2 + m∠3 = 180° Linear pair conjecture
HJJG
HJJG
AC and CB form transversals between parallel lines EC and AB
m∠1 = m∠ 4 and m∠3 = m∠5 because AIA are congruent
Substituting into the first equation above
m∠ 2 + m∠ 4 + m∠5 = 180°
Therefore, the sum of the measures of the angles in every triangle is 180°.
Page 204 #18 Prove Third Angle Conjecture
B
F
Conjecture: If two angles of one triangle are congruent to
two angles of another triangle, then the third angle in each
triangle is congruent to the third angle in the other triangle.
A
Given:
Show:
m∠A = m∠E and m∠B = m∠F
m∠C = m∠D
C
E
D
m∠A + m∠B + m∠C = 180° and m∠E + m∠F + m∠D = 180° by the
triangle sum conjecture.
Since they both equal 180, m∠A + m∠B + m∠C = m∠ E + m∠F + m∠D
Now subtract equal measures m∠A = m∠E and m∠B = m∠F .
m∠C = m∠D Therefore, the third angles are always congruent.
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4.1 Page 203 Exercise #8
a = 69, b = 47, c = 116,
d = 93, e = 86
4.1 Page 203 Exercise #9
m = 30, n = 50, p = 82, q = 28,
r = 32, s = 78, t = 118, u = 50
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4.2 Page 209 Exercise #10
a = 124, b = 56, c = 56,
d = 38, e = 38, f = 76,
g = 66, h = 104, k = 76,
n = 86, p = 38
4.2 Page 209 Exercise #11
a = 36, b = 36,
c = 72, d = 108,
e = 36
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4.3 Page 219 Exercise #17 The Triangle Exterior Angle Conjecture
Conjecture: The measure of an exterior angle of a
triangle is equal to the sum of the measures of the remote
interior angles.
Given: ΔABC with exterior angle
angles labeled as shown.
Show:
∠BCD and
x = a+b
x + c = 180 Linear pair conjecture
a + b + c = 180 Triangle Sum conjecture
Since they both equal 180, x + c = a + b + c
now subtract measure c from both sides
x = a + b Therefore, the sum of the measures of the remote interior angles
equals the measure of an exterior angle.
4.4 Page 226 Exercise #23
a = 37, b = 143, c = 37, d = 58
e = 37, f = 53, g = 48, h = 84,
k = 96, m = 26, p = 69, r = 111,
s = 69
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EXERCISES Lesson 4.4 Page 224-225 #1 – 17
Mark diagrams! If congruence cannot be determined, draw a counterexample.
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EXERCISES Lesson 4.5 Page 229-230 #1 – 18
Mark diagrams! If congruence cannot be determined, draw a counterexample.
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4.6 Page 232 Example A
Given:
Prove:
AM ≅ MB and m∠A = m∠B
AD ≅ BC
C
B
2
1
AM ≅ MB and m∠A = m∠B given
m∠1 = m∠ 2 Vertical angles are congruent.
ΔAMD ≅ ΔBMC by ASA Congruence
AD ≅ BC by CPCTC
4.6 Page 232 Example B
Given: ∠ ADF ≅ ∠ BEF and
Prove:
A
M
D
∠FAB ≅ ∠FBA
AE ≅ BD
∠ADF ≅ ∠BEF and ∠FAB ≅ ∠FBA given
AB ≅ AB Same segment
ΔADB ≅ ΔBEA by AAS Congruence
AE ≅ BD by CPCTC
4.6 Page 234 Exercise #18
a = 112, b = 68, c = 44, d = 44
e = 136, f = 68, g = 68, h = 56,
k = 68, l = 56, m = 124
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EXERCISES Lesson 4.7 Page 239-240 #1 – 5
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4.7 Page 241 Exercise #13
a = 72, b = 36, c = 144, d = 36
e = 144, f = 18, g = 162, h = 144,
j = 36, k = 54, m = 126
4.8 Page 247 Exercise #12
a = 128, b = 128, c = 52, d = 76
e = 104, f = 104, g = 76, h = 52,
j = 70, k = 70, l = 40, m = 110,
n = 58
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EXERCISES Lesson 4.8 Page 246 #4 – 5
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EXERCISES Ch 4 Review Page 252 #7 – 24
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4.R Page 253 Exercise #27
a=?
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