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This paper focuses on the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>K</mi><mi>i</mi></msub></semantics></math></inline-formula>-groups of two types of extensions of abelian categories, which are the trivial extension and the gluing of abelian categories. We prove that, under some conditions, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>K</mi><mi>i</mi></msub></semantics></math></inline-formula>-groups of a certian subcategory of the trivial extension category is isomorphic to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>K</mi><mi>i</mi></msub></semantics></math></inline-formula>-groups of the similar subcategory of the original category. Moreover, under some conditions, we show that the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>K</mi><mi>i</mi></msub></semantics></math></inline-formula>-groups of a left (right) gluing of two abelian categories are isomorphic to the direct sum of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>K</mi><mi>i</mi></msub></semantics></math></inline-formula>-groups of two abelian categories. As their applications, we obtain some results of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>K</mi><mi>i</mi></msub></semantics></math></inline-formula>-groups of the trivial extension of a ring by a bimodule (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow></semantics></math></inline-formula>).