7.2.1 – 7.2.6
By tracing and reflecting triangles to form quadrilaterals, students discover properties
about quadrilaterals. More importantly, they develop a method to prove that what they
have observed is true. Students are already familiar with using flowcharts to organize
information, so they will use flowcharts to present proofs. Since they developed their
conjectures by reflecting triangles, their proofs will rely heavily on the triangle
congruence conjectures developed in Chapter 6. Once students prove that their
observations are true, they can use the information in later problems.
See the Math Notes boxes in Lessons 7.2.1, 7.2.3, 7.2.4, and 7.2.6.
Example 1
ABCD at right is a parallelogram. Use this fact and
A
B
other properties and conjectures to prove that:
a.
the opposite sides are congruent.
b.
the opposite angles are congruent.
C
D
c.
the diagonals bisect each other.
A
B
Because ABCD is a parallelogram, the opposite sides are parallel.
Whenever we have parallel lines, we should be looking for some
pairs of congruent angles. In this case, since AB || CD,
∠BAC ≅ ∠DCA because alternate interior angles are congruent.
Similarly, since AD || CB, ∠DAC ≅ ∠BCA. Also, AC ≅ CA by the
C
D
Reflexive Property. Putting all three of these pieces of information
together tells us that ΔBAC ≅ ΔDCA by the ASA ≅ conjecture. Now that we know that the
triangles are congruent, all the other corresponding parts are also congruent. In particular,
AB ≅ CD and AD ≅ CB , which proves that the opposite sides are congruent. As a flowchart
proof, this argument would be presented as shown below.
Def. of
parallelogram
ABCD is a parallelogram
Reflexive
∠DAC ≅ ∠BCA
Given
prop.
Def. of
Alt. int.
parallelogram
∠BAC ≅ ∠DCA
ΔBAC ≅ ΔDCA
ASA
Alt. int.
≅ Δs
≅ parts
Parent Guide with Extra Practice
97