More on Quadrilaterals and Proofs.
With your group decide which letter of the appropriate polygon(s) that possess each
characteristic. There may be more than one correct letter for each blank.
A. Parallelogram
B. Quadrilateral
C. Kite
D. Rhombus
E. Square
F. Rectangle
1) ________________________ Diagonals are perpendicular.
2) ________________________ Diagonals bisect each other.
3) ________________________ All angles are 90°.
4) ________________________ Both pairs of opposite sides are parallel.
5) ________________________ Each diagonal bisects two angles.
6) ________________________ Diagonals are congruent.
7) ________________________ The diagonals divide the quadrilateral into two
congruent triangles.
8) ________________________ Both pairs of opposite sides are congruent.
Be sure that all the members of your group have their “Theorem Toolkit” up to date with
the information about all of the quadrilaterals.
In Section 7.2.6 there are two different types of theorems we should include in the toolkit.
Two refer to Trapezoids, a quadrilateral with at least one pair of parallel sides and one
refers to all triangles.
For a quadrilateral with two consecutive right angles: If a quadrilateral has two
consecutive right angles, it is a trapezoid.
And for Iscoseles trapezoids, trapezoids with congruent base angles:
If a trapezoid is isosceles, then it has a pair of opposite equal sides.
The final theorem is the Triangle Midsegment Theorem
A midsegment of a triangle is a segment that connects the midpoints of any two sides of
a triangle. Every triangle has three midsegments, as shown below.
A midsegment between two sides of a triangle is half the length of
and parallel to the third side of the triangle. For example, in
ΔABC above,
is a midsegment,
, and DE =
AC.
We will explore this theorem more in the coming days.
More Proofs: Using Triangle congruence and the Properties of Quadrilaterals.
For the following problems write either a Flow-Proof or Two-Column Proof for each of
the exercises. The given information and the diagram is your teams starting point and the
prove statement is the target.
1.
Given: BD and AE bisect each other
Prove: A  E
B
C
A
E
D
2.
D
E
Given: AE BD; D  E
B is the midpoint of AC
Prove: EB  DC
A
B
C
3.
Given: AE BD; AE @ BD
E
A
Prove: DACE @ DDCB
C
B
D
4)
Fill in the missing steps in the proof below.
Given: ABCD is a parallelogram
Prove: DABE @ DCDE
Statements
Reasons
1)
ABCD is a parallelogram
1)__________________
2)
AB = DC
2)__________________
3)
AE = CE
3)__________________
4)
BE = DE
4)__________________
5)
____________________
5)__________________