Question

Given: ∠KJL and ∠KML intercept arc KL.

Prove: m∠KJL = m∠KML

A circle is shown. Angles K J L and K M L intercept arc K L. Angles J K M and J L M intercept arc J M.
♣ =

♦ =

♠ =

Answer 1

Reason: Inscribed-angle theorem — an inscribed angle measures half the measure of its intercepted arc. So any two inscribed angles that intercept the same arc have equal measures.

1) Both ∠KJL and ∠KML intercept arc KL, so
m∠KJL = 1/2 m(arc KL) = m∠KML.
So ∠KJL ≅ ∠KML.

2) Both ∠JKM and ∠JLM intercept arc JM, so
m∠JKM = 1/2 m(arc JM) = m∠JLM.
So ∠JKM ≅ ∠JLM.

Filling the symbols:
- ♣ = ∠KJL = ∠KML
- ♦ = ∠JKM = ∠JLM
- ♠ = m∠KJL = 1/2·m(arc KL) (equivalently m∠KML = 1/2·m(arc KL))