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I can do:

Region[Line[{{0, 0}, {1, 0}}]]

And I can do:

Region[Circle[{1, 0.5`}, 0.5`, {-90 \[Degree], 90 \[Degree]}]]

Can I combine those into one region? Something like:

Region[{Line[{{0, 0}, {1, 0}}] ,Circle[{1, 0.5`}, 0.5`, {-90 \[Degree], 90 \[Degree]}]}]
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RegionUnion gives the union of the regions

r1 = Region[Line[{{0, 0}, {1, 0}}]];
r2 = Region[Circle[{1, 1/2}, 1/2, {-π/2, π/2}]];

RegionUnion[r1, r2]

enter image description here

From Mathematica 10 onwards there is a set of function for Derived Geometric Regions providing several ways of deriving new regions from existing ones, including combining them through Boolean operations and transforming them through a mapping.

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