Question

Question
What are the missing parts that correctly complete the proof?

Given: Point M is on the bisector of angle J K L. Prove: Point M is equidistant from the sides of angle J K L. Art: Three rays K J, K M, and K L point in rightward direction sharing a common endpoint at K. Ray K M is horizontal and bisects angle J K L equally. The perpendicular from point M to ray K J is made by a dotted line. The perpendicular cuts ray K J at X. Angle K X M is labeled a right angle. The perpendicular from point M to ray K L is made by another dotted line. The perpendicular cuts ray K L at Y. Angle K Y M is labeled a right angle.

Drag the answers into the boxes to correctly complete the proof.

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Statement Reason
​KM−→−
​ is the bisector of ∠JKL
. Given
Press enter key to drop response here.Response input area Press enter key to drop response here.Response input area
∠KXM
and ∠KYM
are right angles. Given
​∠KXM≅∠KYM
​ All right angles are congruent.
Press enter key to drop response here.Response input area Reflexive Property of Congruence
△KXM≅△KYM
Press enter key to drop response here.Response input area
​MX¯¯¯¯¯¯≅MY¯¯¯¯¯¯
​ Press enter key to drop response here.Response input area
Point M is equidistant from the sides of ∠JKL
. Definition of equidistant

Answer 1

Fill in the blanks as follows:

- ∠XKM ≅ ∠MKY — Reason: Definition of angle bisector (KM bisects ∠JKL)
- KM ≅ KM — Reason: Reflexive Property of Congruence
- △KXM ≅ △KYM — Reason: ASA (Angle–Side–Angle)
- MX ≅ MY — Reason: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

This yields MX = MY, so M is equidistant from the sides of ∠JKL.