# 9.07 Justifying constructions using the properties of quadrilaterals

Lesson

We can use a set of circular arcs to construct geometric figures like quadrilaterals. We can use the property of quadrilaterals to go one step further and construct bisectors, angle bisectors, or a pair of perpendicular or parallel lines. For instance, the following pair of perpendicular line segments are the diagonals of the rhombus $ACBD$ACBD.

 $\overline{AB}$AB is perpendicular to $\overline{DC}$DC

So given a set of arcs, we want to identify what quadrilateral was made in order to prove geometric statements.

#### Exploration

Let's determine which order the arcs were drawn. From the image above, we can see that we are given the point $A$A, and that the arcs below were drawn first.

 First, arcs drawn with center at $A$A

This allows us to place the point $B$B along the middle arc, and create the second pair of arcs with same radius to create the points $C$C and $D$D.

 Then, arcs were drawn centered at $B$B

Then the perpendicular lines were drawn.

 $\overline{AB}$AB is perpendicular to $\overline{DC}$DC

To prove that $\overline{AB}$AB is perpendicular to $\overline{DC}$DC, we first notice that the sides $\overline{AC}$AC, $\overline{CB}$CB, $\overline{BD}$BD, and $\overline{DA}$DA are all congruent since they've been constructed by the same radius. So by definition, this means that $ACBD$ACBD is a rhombus. A special property of a rhombus is that the diagonals are perpendicular. In this case, $\overline{AB}$AB is perpendicular to $\overline{DC}$DC.

We can formalize the above explanation in a two column proof as shown below.

 Statements Reasons $\overline{AC}\cong\overline{CB}\cong\overline{BD}\cong\overline{DA}$AC≅CB≅BD≅DA The arcs that created the segments have the same radius. $ACBD$ACBD is a rhombus Definition of a rhombus $\overline{AB}\cong\overline{DC}$AB≅DC If a parallelogram is a rhombus, then its diagonals are perpendicular.

#### Practice questions

##### question 1

In the image below, the line $\overleftrightarrow{AB}$AB is constructed parallel to the given line $\overleftrightarrow{DC}$DC. Justify the steps of construction in a two column proof.

##### question 2

In the image below, $\overline{PR}$PR is constructed so that $\angle TPR$TPR is congruent to $\angle RPQ$RPQ. Justify the steps of construction in a two column proof.

### Outcomes

#### GEO-G.CO.12

Make, justify, and apply formal geometric constructions.