Question

What are the missing parts that correctly complete the proof? given: point p is on the bisector of angle a b c. prove: point p is equidistant from the sides of angle a b c. art: three rays b a, b p, and b c point in rightward direction sharing a common endpoint labeled as b. ray b p is horizontal and bisects angle a b c equally. the perpendicular from point p to ray b a is made by a dotted line. the perpendicular cuts ray b a at point x. angle b x p is labeled a right angle. the perpendicular from point p to ray b c is made by another dotted line. the perpendicular cuts ray b c at point y. angle b y p is labeled a right angle. drag the answers into the boxes to correctly complete the proof. put responses in the correct input to answer the question. select a response, navigate to the desired input and insert the response. responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. responses can also be moved by dragging with a mouse. statement reason

1. bp−→− is the bisector of ∠abc. given
2. ∠abp≅∠cbp
3.∠bxp and ∠byp are right angles. given
4. all right angles are congruent.
5. bp¯¯¯¯¯≅bp¯¯¯¯¯
6. △bxp≅△byp
7. corresponding parts of congruent triangles are congruent.
8. point p is equidistant from the sides of ∠abc. definition of equidistant

Answer 1

To prove that point P is equidistant from the sides of angle ABC, we must complete the proof by using definitions related to angle bisectors, properties of congruent triangles, and the concept of equidistance. After establishing the congruence of angles and sides involved, we conclude that PX is equal to PY, proving P's equidistance.

Step-by-step explanation:

To complete the proof that point P is equidistant from the sides of ∆ABC when P is on the bisector of angle ABC, we fill in the proof as follows:

Point P is on the bisector of ∠ABC. (Given)

∠ABP ≅ ∠CBP. (Definition of an angle bisector)

∠BXP and ∠BYP are right angles. (Given)

All right angles are congruent. (Definition of congruent angles)

─BP ≅ ─BP. (Reflexive Property of congruence)

∆BXP ≅ ∆BYP. (Angle-Side-Angle postulate)

PX ≅ PY. (Corresponding Parts of Congruent Triangles are Congruent (CPCTC))

Point P is equidistant from the sides of ∠ABC. (Definition of equidistant)

This proof relies on the properties of congruent triangles and the definitions of angle bisectors and equidistance to establish that PX ≅ PY, thereby showing P is equidistant from the sides of angle ABC.