Using Parallelograms
Proofs Using Congruent Triangles –The majority of the proofs in this section use congruent triangles. The
quadrilateral of interest is somehow divided into triangles that can be proved congruent with the theorems and
postulates of the previous chapters. Once the triangles are known to be congruent, the definition of congruent
triangles ensures that certain parts of the quadrilateral are also congruent. Students should be made aware of this
pattern if they are having difficulty writing or understanding the proofs of the properties of various quadrilaterals. If
they are still struggling they should spend some time reviewing sections two through six of Chapter Four: Congruent
Triangles.
The Diagonals of Parallelograms –The properties concerning the sides and angles of parallelograms are fairly
intuitive, and students pick them up quickly. More emphasis should be placed on what is known, and not known
about the diagonals. Students frequently try to use the incorrect fact that the diagonally of a parallelogram are
congruent. Rectangles are the focus of an upcoming lesson, but demonstrating to students that the diagonals of a
quadrilateral are only congruent in the special case where all the angles of the parallelogram are congruent. For a
general parallelogram, the measures of the two pieces of the same diagonal separated by the other diagonal, can be
set equal to each other, but no comparison can be made between diagonals.
Additional Exercises:
- QuadrilateralABCDis a parallelogram.
AB= 2 x+ 5 ,BC=x−3, andDC= 3 x− 10
Find the measures of all four sides of the quadrilateral.
(Hint: Draw and label a picture. Remember the name of a polygon lists the vertices in a circular order.)
Answer:
2 x+ 5 = 3 x− 10 AB=CD=35 andBC=AD= 12
x= 15- JACK is a parallelogram.
m^6 A− 10 x− 60 ◦andm^6 C= 2 x+ 45 ◦
Find the measures of all four angles.
Answer:
10 x− 60 + 2 x+ 45 = 180 m^6 C=m^6 J= 77. 5 ◦x= 161
4
m^6 A=m^6 K= 102. 5 ◦Proving Quadrilaterals are Parallelograms
Proof Practice –The proofs in this section may seem a bit repetitive, but students will benefit from practicing these
proofs since they review important concepts learned earlier in the course. To avoid loosing the students’ attention,
find different ways of presenting the proofs. One idea is to divide the students into groups, and have each group
demonstrate a different proof to the class.
Chapter 2. Geometry TE - Common Errors