CK-12 Geometry Concepts

http://www.ck12.org Chapter 6. Polygons and Quadrilaterals

4. Draw the diagonals. Measure each and then measure the lengths from the point of intersection to each vertex.

To continue to explore the properties of a parallelogram, see the website:

http://www.mathwarehouse.com/geometry/quadrilaterals/parallelograms/interactive-parallelogram.php

In the above investigation, we drew a parallelogram. From this investigation we can conclude:

Opposite Sides Theorem:If a quadrilateral is a parallelogram, then the opposite sides are congruent.

Opposite Angles Theorem:If a quadrilateral is a parallelogram, then the opposite angles are congruent.

Consecutive Angles Theorem:If a quadrilateral is a parallelogram, then the consecutive angles are supplementary.

Parallelogram Diagonals Theorem:If a quadrilateral is a parallelogram, then the diagonals bisect each other.

To prove the first three theorems, one of the diagonals must be added to the figure and then the two triangles can be
proved congruent.

Proof of Opposite Sides Theorem:

Given:ABCDis a parallelogram with diagonalBD

Prove:AB∼=DC,AD∼=BC

TABLE6.4:

Statement Reason

1.ABCDis a parallelogram with diagonalBD Given

2.AB||DC,AD||BC Definition of a parallelogram

3.^6 ABD∼=BDC,^6 ADB∼=DBC Alternate Interior Angles Theorem
4.DB∼=DB Reflexive PoC
5. 4 ABD∼= 4 CDB ASA
6.AB∼=DC,AD∼=BC CPCTC

Example A

ABCDis a parallelogram. Ifm^6 A= 56 ◦, find the measure of the other three angles.

Draw a picture. When labeling the vertices, the letters are listed, in order, clockwise.

Ifm^6 A= 56 ◦, thenm^6 C= 56 ◦because they are opposite angles.^6 Band^6 Dare consecutive angles with^6 A, so
they are both supplementary to^6 A.m^6 A+m^6 B= 180 ◦, 56 ◦+m^6 B= 180 ◦,m^6 B= 124 ◦.m^6 D= 124 ◦.

Example B

Find the values ofxandy.

Opposite sides are congruent, so we can set each pair equal to each other and solve both equations.

6 x− 7 = 2 x+ 9 y^2 + 3 = 12

4 x= 16 y^2 = 9

x= 4 y= 3 or− 3

Even thoughy=3 or -3, lengths cannot be negative, soy=3.