[[!redirects two triangles in a square solutions]] # Solution to the [[Two Triangles in a Square]] Puzzle +-- {.image} [[TwoTrianglesinaSquare.jpeg:pic]] > The outer shape is a square. What’s the missing angle? =-- ## Solution by [[Square|Symmetries of a Square]], [[Angles in a Triangle]], and [[Angles at a Point on a Straight Line]] +-- {.image} [[TwoTrianglesinaSquareLabelled.jpeg:pic]] =-- Label the points as above. Reflecting the square in the diagonal $B F$ swaps $A$ and $D$ and leaves $E$ where it is. Therefore, angles $B \hat{A} E$ and $E \hat{D} B$ are equal. Let us write this angle as $a^\circ$. Then as [[angles in a triangle]] add up to $180^\circ$, and [[angles at a point on a straight line]] also add up to $180^\circ$, angle $A \hat{C} B$ is equal to $2 a^\circ$. As triangle $A B C$ is [[right-angled triangle|right angled]], angles $A \hat{C} B$ and $B \hat{A} C$ add up to $90^\circ$, so $3 a^\circ = 90^\circ$ meaning that $a^\circ = 30^\circ$. Therefore, angle $B \hat{A} C = 30^\circ$.