$44$ . If triangles $A B C$ and $D E C$ are both equilateral, show that $D E \| A B$ .

Full solution

44

. If triangles

A B C

and

D E C

are both equilateral, show that

D E \| A B

Equilateral Triangles Definition: Given that triangles $ABC$ and $DEC$ are both equilateral, we know that all their internal angles are equal to $60$ degrees.
Angle Properties: In an equilateral triangle, all sides are of equal length, and all angles are equal. Therefore, angle $ABC$ in triangle $ABC$ is $60$ degrees, and angle $DEC$ in triangle $DEC$ is also $60$ degrees.
Linear Pair Angles: Since triangle $ABC$ is equilateral, angle $CAB$ is also $60$ degrees. Similarly, angle $CDE$ in triangle $DEC$ is $60$ degrees.
Supplementary Angles: Now, let's consider the straight line formed by points $A$ , $C$ , and $E$ . Since angles $CAB$ and $CDE$ are both $60$ degrees, and they are on the same straight line, they form a linear pair which adds up to $180$ degrees.
Alternate Interior Angles Theorem: The fact that angles $CAB$ and $CDE$ add up to $180$ degrees means that they are supplementary. Since they are equal (both are $60$ degrees), the lines $AB$ and $DE$ are parallel to each other by the alternate interior angles theorem.
Parallel Lines Conclusion: To summarize, we have shown that angle $ABC$ is equal to angle $DEC$ , and angle $CAB$ is equal to angle $CDE$ . Since these angles are alternate interior angles and are equal, lines $AB$ and $DE$ are parallel by definition.