Chapter 13 (PDF)

4. Given: AC and BD bisect each
other at M and AC BD
Show: ABCD is a rhombus
Flowchart Proof
A
(See flowchart at bottom of page.)
5. Given: ABCD is a
trapezoid with AB CD
and A B
Show: ABCD is isosceles
D C
Proof:
A
E
B
Statement Reason
1. ABCD is a trapezoid
with AB CD
1. Given
2. Construct CE AD 2. Parallel Postulate
3. AECD is a 3. Definition of
parallelogram parallelogram
4. AD CE 4. Opposite Sides
Congruent Theorem
5. A BEC 5. CA Postulate
6. A B 6. Given
7. BEC B 7. Transitivity
8. ECB is isosceles 8. Converse of IT
Theorem
9. EC CB 9. Definition of isosceles
triangle
10. AD CB 10. Transitivity
11. ABCD is isosceles 11. Definition of isosceles
trapezoid
Lesson 13.4, Exercise 4
AC and BD bisect
each other at M
Given
AC BD
Given
ABCD is a
parallelogram
Converse of the
Parallelogram
Diagonals Theorem
DM BM
Definition of bisect,
definition of
congruence
DMA and BMA
are right angles
Definition of
perpendicular
D C
M
B
AB DC
Opposite Sides
Theorem
AD BC
Opposite Sides
Theorem
AM AM
Reflexive property
DMA BMA
Right Angles
Congruent Theorem
6. Given: ABCD is a
trapezoid with AB CD
and AC BD
Show: ABCD is isosceles
Proof:
Statement Reason
1. ABCD is a trapezoid
with AB CD
1. Given
2. Construct BE AC 2. Parallel Postulate
3. DC and BE intersect 3. Line Intersection
at F Postulate
4. ABFC is a 4. Definition of
parallelogram parallelogram
5. AC BF 5. Opposite Sides
Congruent Theorem
6. AC BD 6. Given
7. BF BD 7. Transitivity
8. DFB is isosceles 8. Definition of isosceles
triangle
9. DFB FDB 9. IT Theorem
10. CAB DFB 10. Opposite Angles
Theorem
11. FDB DBA 11. AIA Theorem
12. CAB DBA 12. Transitivity
13. AB AB 13. Reflexive property
14. ACB BDA 14. SAS Postulate
15. AD BC 15. CPCTC
16. ABCD is isosceles 16. Definition of isosceles
trapezoid
ADM ABM
SAS Postulate
All 4 sides are
congruent
Transitivity
AD AB
CPCTC
D
C
A B
ABCD is a
rhombus
Definition of
rhombus
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