of a rhombus bisect the angles of a rhombus."
A Proof Outline
by David Wise
Note: I recommend that this page be printed out, so that the
instructions are easier to follow.
In order to successfully complete a proof,
it is important to think of the definition and the construction
of a rhombus.
In the following outline, I will provide the statements, you
provide the reasons.
Prove: If a quadrilateral is a rhombus, then
the diagonals bisect the angles.
Given: Rhombus ABCD with diagonals BD and AC.
Prove: Segment AC bisects angles
BAD and DCB. Segment DB bisects angles ADC and CBA.
- A rhombus is a parallelogram, so the definition
and properties of a parallelogram apply to a rhombus.
- Consider how a rhombus is constructed------parallel
- Consider properties of parallel lines and
- Each diagonal creates a pair of triangles.
- Consider triangle congruency properties.
- Consider isosceles triangles and their properties.
- Click here to investigate this sketch to help with the steps
of the proof.
First we will use only diagonal AC.
- Segment DA is congruent to segment AB.
- Segment CD is congruent to segment CB.
- Segment CA is congruent to segment CA.
- Triangle DAC is congruent to triangle BAC.
- Angle CAD is congruent to angle BAC. Angle
DCA is congruent to angle ACB.
- Segment AC bisects angles BAD
Then we will use only diagonal BD.
- Segment DA is congruent to segment DC.
- Segment AB is congruent to segment BC.
- Segment BD is congruent to segment BD.
- Triangle DAB is congruent to triangle DCB.
- Angle ADB is congruent to angle BDC. Angle
DBA is congruent to angle CBD.
- Segment BD bisects angles ADC
- Transform the two-column
proof into a paragraph proof.
- Find an alternative way to
prove that the diagonals of a parallelogram bisect each other.
If you have any questions while trying to complete this investigation,
or suggestions that would be useful, especially for use at the
high school level, please send e-mail to firstname.lastname@example.org.
the Table of Contents.