Question

Given: EL≅KP Prove: KE≅LP

Answer 1

See Below.

Explanation:

We are given that:

`EL\cong KP`

And we want to prove that:

`KE\cong LP`

Congruent chords have congruent arcs. Therefore:

`\stackrel{\frown}{EL}\, \cong \, \stackrel{\frown}{KP}`

Arc EL is the sum of Arcs LP and PE:

`\stackrel{\frown}{EL}\,=\, \stackrel{\frown}{LP}+\stackrel{\frown}{PE}`

Likewise, Arc KP is the sum of Arcs KE and PE:

`\stackrel{\frown}{KP}\, =\, \stackrel{\frown}{KE}+\stackrel{\frown}{PE}`

Since Arcs EL and KP are congruent:

`\stackrel{\frown}{LP}+\stackrel{\frown}{PE}\, =\, \stackrel{\frown}{KE}+\stackrel{\frown}{PE}`

Subtraction Property of Equality:

`\stackrel{\frown}{LP} \, \cong\, \stackrel{\frown}{KE}`

Congruent arcs have congruent chords. Therefore:

`LP\cong KE`