5.08 Justifying constructions using congruent triangles

Geometric constructions can be made by first drawing a set of circular arcs. These arcs constitute points of interest such as vertices, from which we can join to create a figure like an angle or a triangle.

Construction of isosceles triangle $ABC$ABC

 

We want to be able to interpret how a construction was formed given a set of circular arcs. In the above image, the dotted arc comes from a circle centered at $A$A which passes two arbitrarily chosen points, $B$B and $C$C. The constructed triangle $\triangle ABC$△ABC is isosceles, since the line segments $\overline{AB}$AB and $\overline{AC}$AC are congruent radii of the same circle.

 

Exploration

In a diagram, we assume that every arc belongs to a circle that is centered at a given or constructed point. In the following image, arc $1$1 comes from a circle centered at the given point $C$C while arcs $2$2 and $3$3 come from a circle centered at the constructed point $A$A.

Construction of arcs and vertices

 

We can also identify the order that the arcs were drawn. In the image above, arc $1$1 must be created first to construct the point $A$A. Then the arcs $2$2 and $3$3 can be drawn second, since they center at $A$A.

We lastly assume that certain arcs have equal radii. For instance, the arcs that form $A$A and $B$B share the same radii. Also, the four arcs at $D$D and $E$E all share the same radii.

Identifying when arcs have equal radii is important, since it allows us to make geometric statements about our constructions. For instance, the line $\overleftrightarrow{AB}$›‹AB bisects the line segment $\overline{DE}$DE. We can show this using congruent triangles.

$\overleftrightarrow{AB}$›‹AB bisects $\overline{DE}$DE

 

To prove that $\overleftrightarrow{AB}$›‹AB bisects $\overline{DE}$DE, we first want to show that the triangles $\triangle ADB$△ADB and $\triangle AEB$△AEB are congruent.

Note that $\overline{AD}$AD is congruent to $\overline{AE}$AE and $\overline{DB}$DB is congruent to $\overline{EB}$EB because the arcs that created these congruent segments have the same radii. Of course $\overline{AB}$AB is a common side to both triangles, and so $\triangle ADB$△ADB is congruent to $\triangle AEB$△AEB by side-side-side congruence.

The next step is to show that $\triangle ADC$△ADC and $\triangle AEC$△AEC are congruent.

We already have a pair of sides that are congruent from before, $\overline{AD}$AD and $\overline{AE}$AE. $\overline{AC}$AC is a common side to both triangles, and so is clearly congruent to itself. Lastly we know that $\angle DAC$∠DAC is congruent to $\angle EAC$∠EAC because they represent corresponding angles from the pair of congruent triangles, $\triangle ADC$△ADC and $\triangle AEC$△AEC. So $\triangle ADC$△ADC and $\triangle AEC$△AEC have side-angle-side congruence.

This means that $\overline{DC}$DC is congruent to $\overline{EC}$EC because they are corresponding sides of the congruent triangles, $\triangle ADC$△ADC and $\triangle AEC$△AEC. This means that $\overleftrightarrow{AB}$›‹AB must bisect $\overline{DE}$DE.

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

Statements	Reasons
$\overline{AD}\cong\overline{AE}\cong\overline{DB}\cong\overline{EB}$AD≅AE≅DB≅EB	The arcs that created the segments have the same radius.
$\overline{AB}\cong\overline{AB}$AB≅AB	Reflexive property of congruence
$\triangle ADB\cong\triangle AEB$△ADB≅△AEB	Side-side-side congruence of triangles
$\overline{AC}\cong\overline{AC}$AC≅AC	Reflexive property of congruence
$\angle DAC\cong\angle EAC$∠DAC≅∠EAC	Corresponding parts of congruent triangles are congruent (CPCTC).
$\triangle ADC\cong\triangle AEC$△ADC≅△AEC	Side-angle-side congruence of triangles
$\overline{DC}\cong\overline{EC}$DC≅EC	Corresponding parts of congruent triangles are congruent (CPCTC).
$\overleftrightarrow{AB}$›‹AB bisects $\overline{DE}$DE	Definition of bisector

 

Practice questions

question 1

question 2