Chapter 5 - Frost Middle School

Chapter 5
8.
Proofs Using Congruence
Lesson 5-4
(pp. 269–276)
Mental Math
Conclusions
1. m n
2. ∠1 ∠5
3. ∠5 ∠8
a. reflection over a horizontal line
b. reflection over a vertical line
4. ∠1 ∠8
c. the letter “d”
Justifications
Given
Corresponding Angles
Postulate
Vertical Angles
Theorem
Transitive Property
of Congruence
9. The sides of the handicapped space one parallel
to the sides of the regular space. Therefore the
angles are corresponding angles formed by
parallel lines. So by the Parallel Lines Postulate,
they are the same measures.
Activity
10. ∠2 = ∠3 = ∠6 = ∠7 = 43,
∠1 = ∠4 = ∠5 = ∠8 = 137
11. Answers vary. Sample:
Guided Example 2
Vertical Angles Theorem; Transitivity Property of
Congruence
Questions
1. Thales was a Greek mathematician who lived in
the 6th century BCE. He is the first person to
write proofs like those used today.
2. the construction of an equilateral triangle from
two overlapping circles
3. All points on a circle are equidistant from the
center of the circle by the definition of a circle.
4. given, prove, drawing, and proof
5. Conclusions
1. A with radius AB
2. AD = AB
3. B with radius BA
4. AB = BD
5. AD = BD
6. ABD is equilateral.
Justifications
Given
definition of circle
Given
definition of circle
Transitive Property of
Equality
definition of
equilateral triangle
6. a. ∠3, ∠4, ∠5, ∠6
b. ∠3, ∠5; ∠4, ∠6
c. m and n are not necessarily parallel.
7. a. true
b. false
A80
Geometry
12. a. congruent; corresponding angles are
congruent
b. congruent; alternate interior angles are
congruent
c. supplementary; same-side interior angles are
supplementary
13. a. ∠C and ∠E, ∠D and ∠G
___
21. T(C) = C by the A-B-C-D Theorem.
___
b. CG and DE
22. B
___
14. a. ∠1 and ∠4, ∠3 and ∠5
___
23. a. m is the perpendicular bisector of PQ.
___
b. AB and PQ
b. A lies on m.
15. Conclusions
___
1. B is the midpoint of ___
AC;
C
is
the
midpoint
of
BC
___
___
___
___
2. ___
AB ___
BC and BC CD
3. AB CD
Justifications
Given
def. of midpoint
Trans. Prop. of 16. ∠1 and ∠2 are supplementary; m∠2 = x; 180;
180; a linear pair; 180; m∠1 = m∠3; congruent;
Corresponding Angles; if two lines are cut by a
transversal and form congruent corresponding
angles, the two lines are parallel.
17. Refer to the diagram in Question 16.
Given: m n
Prove: ∠1 and ∠2 are supplementary angles.
Proof: Since ∠2 and ∠3 are a linear pair, they are
supplementary angles. So m∠3 + m∠2 = 180.
Since ∠1 and ∠3 are corresponding angles,
m∠1 = m∠3. By substitution, m∠1 + m∠2 = 180,
and thus ∠1 and ∠2 are supplementary angles.
18. a.
A
B
F
D
C
E
b. ___
Answers
___vary. Sample: ∠DAB ∠FBC and
AB BC because corresponding parts of
congruent figures must be congruent.
___
___
19. AD LO
20. False; Answers vary. Sample:
___
___
m AB
___ = 1.75 cm
mFE = 1.75 cm
m___
CD = 3.25 cm
mGH = 3.25 cm
E
D
A
B
A81
H
F
C
Geometry
G
___
c. A is on the perpendicular bisector of PQ.
24. a. no
b. No; adding one arc will at most reduce the
number of odd nodes to 3, and thus the
network will still not be traversable.
25. No, this congruence only requires two sides
and two angles of the same length and measure,
respectively.