5.08 Justifying constructions using congruent triangles

Lesson

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

In the image below, $\angle EDF$EDF is constructed congruent to $\angle BAC$BAC. Justify the steps of construction in a two column proof.

question 2

Prove that the construction of $\overrightarrow{AD}$AD is an angle bisector of the given angle $\angle BAC$BAC.

Outcomes

GEO-G.CO.12

Make, justify, and apply formal geometric constructions.

GEO-G.SRT.5a

Use congruence and similarity criteria for triangles to solve problems algebraically and geometrically.

GEO-G.SRT.5b

Use congruence and similarity criteria for triangles to prove relationships in geometric figures.