Use the Isosceles Triangle Theorem to find a pair of congruent angles. Also, remember that all the angles of an equilateral triangle are congruent. Use the definition of supplementary angles.
See solution.
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Since △ HJM is isosceles, we have that JH ≅ MH and, by the Isosceles Triangle Theorem, we have that ∠ J ≅ ∠ M. Additionally, since △ HKL is equilateral, all its angles measure 60^(∘).
We are given that ∠ JKH and ∠ HKL are supplementary, and ∠ HLK and ∠ MLH are supplementary. Then, by the definition of supplementary angles we can write the following equations.
m∠ JKH + m ∠ HKL &= 180^(∘)
m ∠ HLK + m∠ MLH &= 180^(∘)
By substituting m ∠ HKL=60^(∘) = m ∠ HLK, we obtain an important relation.
m∠ JKH + 60^(∘) = 180^(∘) & ⇒ m∠ JKH = 120^(∘)
60^(∘) + m∠ MLH = 180^(∘) & ⇒ m∠ MLH = 120^(∘)
From the above, we get that ∠ JKH ≅ ∠ MLH.
Let's list the congruent parts between △ HJK and △ HML.
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∠ HKJ ≅ ∠ HLM & Angle
∠ J ≅ ∠ M & Angle
JH ≅ MH & Non-included Side
By the Angle-Angle-Side (AAS) Congruence Postulate we obtain that △ HKJ ≅ △ HLM, which implies that ∠ JHK ≅ ∠ MHL.
Paragraph Proof
Given: & △ HJM is isosceles
& △ HKL is equilateral
& ∠ JKH and ∠ HKL are supplementary
& ∠ HLK and ∠ MLH are supplementary
Prove: & ∠ JHK ≅ ∠ MHL
Proof: Since △ HJM is isosceles then JH≅ MH, and by the Isosceles Triangle Theorem we have that ∠ J ≅ ∠ M. We also have that △ HKL is equilateral, which means that ∠ HKL ≅ ∠ HLK. Additionally, we are told that ∠ HKL and ∠ HKJ, and ∠ HLK and ∠ HLM, are supplementary angles.
