Use the definition of an angle bisector and list the congruent parts between both triangles.
See solution.
Practice makes perfect
We want to write a paragraph proof of the following conjecture. Here is also what we know.
Given: & ∠ K ≅ ∠ M, JK ≅ JM
& JL bisects ∠ KLM.
Prove: & △ JKL ≅ △ JML
Let's focus on the diagram. By the definition of an angle bisector, we have that ∠ KLJ and ∠ MLJ are congruent. Let's mark this, as well as other corresponding congruent parts of both triangles, in the diagram.
There are two pairs of congruent angles and a pair of congruent sides in △ JKL and △ JML.
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∠ KLJ ≅ ∠ MLJ & Angle
∠ JKL ≅ ∠ JML & Angle
KJ ≅ MJ & Non-included Side
Two angles and a non-included side of △ JKL are congruent to two angles and a non-included side of △ JML. By the Angle-Angle-Side (AAS) Congruence Postulate we can conclude that the triangles are congruent.
△ JKL ≅ △ JML
Paragraph Proof
We can now summarize our findings in a paragraph proof.
Given: & ∠ K ≅ ∠ M, JK ≅ JM
& JL bisects ∠ KLM
Prove: & △ JKL ≅ △ JML
Proof: By definition of the angle bisector, we have ∠ KLJ ≅ ∠ MLJ. Then, we have that two angles and the non-included side of △ JKL are congruent to the corresponding two angles and side of △ JML. Consequently, by the Angle-Angle-Side (AAS) Congruence Postulate we conclude that △ JKL ≅ △ JML.
