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 â–³ HJK and â–³ 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: & â–³ HJM is isosceles
& â–³ 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.
