The　interaction　between　hydraulic　fractures　(HFs)　and　natural　fractures　(NFs)　has　long　been　a　prominent　focus　in　reservoir　fracturing,　particularly　in　the　development　of　hot　dry　rock　(HDR)　reservoirs.　Activating　existing　NFs　plays　a　crucial　role　in　constructing　a　complex　fracture　network　and　enhancing　reservoir　fracturing　effectiveness.　This　study　utilizes　the　lattice-based　method　XSite　and　the　smooth　joint　models　(SJMs)　to　construct　granite　fracturing　models　with　various　combinations　of　parallel　and　intersecting　NFs　and　explore　the　three-dimensional　(3D)　interaction　behavior　between　fractures.　The　research　findings　demonstrate　that　the　variation　in　the　approach　angle　influences　the　propagation　of　HFs　through　a　pair　of　parallel　NFs,　which　can　occur　in　three　ways:　no　crossing,　offset　crossing,　or　direct　crossing.　As　the　number　of　parallel　NFs　rises,　HFs　predominantly　activate　the　initially　encountered　NFs　on　a　large　scale,　while　the　activation　level　of　subsequent　NFs　markedly　diminishes.　Within　a　cluster　of　intersecting　fractures,　HFs　tend　to　preferentially　pass　through　NFs　with　the　smallest　approach　angle　and　suppress　the　activation　of　NFs　with　higher　approach　angles.　Furthermore,　the　numerical　simulation　results　provide　valuable　insights　into　the　evolution　of　breakdown　pressure,　fracture　aperture,　principal　stress,　lattice　displacement,　and　other　fracturing　outcomes　throughout　the　interaction　process.　These　findings　serve　as　important　references　for　controlling　HF　propagation　and　activating　NFs　in　the　development　of　HDR　reservoirs.