# Solution to the Tilted Triangles Puzzle +-- {.image} [[TiltedTriangles.png:pic]] > The two equilateral triangles are congruent. What's the angle? =-- ## Solution by [[Angle at the Centre is Twice the Angle at the Circumference]] +-- {.image} [[TiltedTrianglesLabelled.png:pic]] =-- With the points labelled as in the above diagram, since the triangles are [[equilateral]] and [[congruent]] the lengths of $O E$, $O D$, $O B$, and $O A$ are all the same so the circle centred on $O$ that passes through $E$ also passes through $D$, $B$, and $A$. Since the [[angle at the centre is twice the angle at the circumference]], the reflex angle $E \hat{O} A$ is twice angle $E \hat{B} A$. The same argument shows that angle $B \hat{O} D$ is twice angle $B \hat{A} D$. Adding together angles $B \hat{A} D$ and $E \hat{B} A$ therefore gives half of the sum of $B \hat{O} D$ and $E \hat{O} A$. Angles $D \hat{O} E$ and $A \hat{O} B$ are both the [[interior angle]] of an [[equilateral triangle]] so are $60^\circ$. Since the [[angles at a point]] add up to $360^\circ$, this means that angles $B \hat{A} D$ and $E \hat{B} A$ add up to $120^\circ$. Then as the [[angles in a triangle]] add up to $180^\circ$, angle $A \hat{C} B$ is $60^\circ$ and so $B \hat{C} D$ is $120^\circ$.