Look for different ways the two triangles can be congruent.
D
d
Look for segments congruent to CB.
E
e
The angle ∠D is one angle of an equilateral triangle.
A
a
Regular hexagons and equilateral triangles
B
b
Example Solution: △ ABC≅ △ EDC
C
c
Example Solution: ∠ABC and ∠EDC
D
d
4, see solution.
E
e
60^(∘), see solution.
Practice makes perfect
a
There are six regular hexagons and twelve equilateral triangles creating the pattern.
b
Any two triangles in the pattern are congruent.
Using the labels given on the figure we can write one of the congruent pairs.
△ ABC≅ △ EDC
Note that since both â–³ ABC and â–³ EDC are regular triangles, any vertex of â–³ ABC can correspond to any vertex of â–³ EDC. This means that the congruence of the same two triangles can be written in six different ways.
△ ABC&≅ △ EDC
△ ABC&≅ △ DCE
△ ABC&≅ △ CED
△ ABC&≅ △ ECD
△ ABC&≅ △ CDE
△ ABC&≅ △ DEC
c
Since both â–³ ABC and â–³ EDC are regular triangles, any angle in â–³ ABC can correspond to any angle in â–³ EDC.
∠ABCand∠EDC
∠ABCand∠DCE
∠ABCand∠CED
∠BCAand∠EDC
∠BCAand∠DCE
∠BCAand∠CED
∠CABand∠EDC
∠CABand∠DCE
∠CABand∠CED
d
There are several segments in this pattern that are congruent to CB. Let's focus on two of these.
Segments CA and CB are congruent, since these are sides of an equilateral triangle.
Segments CE and CB are congruent, since these are sides of a regular hexagon.We know that congruent segments have the same length.
CA=CE=CB=2inches
Point C is between points A and E on a line, so the length of AE is the sum of the lengths of CA and CE.
Angle ∠D is one of the angles of the equilateral triangle △ DEC.
Since â–³ DEC is a regular polygon, it has three congruent angles.
∠D≅∠C≅∠E
Congruent angles have the same measure.
m∠E&=m∠D
m∠C&=m∠D
We can use this and the Triangle Angle-Sum Theorem to find the measure of ∠D.
