In Geometric Algebra, the algebra G(3,0,1) is known as PGA, the plane-based and point-based geometric algebras, or projective geometric algebra, of points, lines, and planes in 3D space. The even-grades subalgebra of PGA, which we call Dual Quaternion Geometric Algebra (DQGA), represents Dual Quaternion Algebra (DQA). In the plane-based algebra of PGA, there are entities for points, lines, and planes and many operations on them, including dualization to the point-based entities, reflections in planes, rotations, translations, projections, rejections, and intersections (meet products). In this paper, we derive a complete set of identities that relate all of the plane-based entities and operations in PGA to their corresponding entities and operations in DQGA. Therefore, this paper contributes into the literature on dual quaternions and PGA the complete details on how to use DQA or DQGA as a geometric algebra of points, lines, and planes with many useful operations. All DQGA entities and operations are defined or derived such that the orientations of the entities are maintained correctly through all of the operations. We also define three new part operators for taking the point, line, or plane part of a dual quaternion, which may improve the computational efficiency of intersection (meet) operations. Dual quaternions already have some applications in computer graphics and kinematics. This paper expands on the understanding of dual quaternions and introduces DQA as a versatile geometric algebra of points, lines, and planes with many new operations that do not appear in prior literature, expanding the possible applications of dual quaternions.