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Complete the paragraph proof. Given: M is the midpoint of PK PK ⊥ MB Prove: △PKB is isosceles It is given that M is the midpoint of PK and PK ⊥ MB. Midpoints divide a segment into two congruent segments, so PM ≅ KM. Since PK ⊥ MB and perpendicular lines intersect at right angles, ∠PMB and ∠KMB are right angles. Right angles are congruent, so ∠PMB ≅ ∠KMB. The triangles share MB, and the reflexive property justifies that MB ≅ MB. Therefore, △PMB ≅ △KMB by the SAS congruence theorem. Thus, BP ≅ BK because . Finally, △PKB is isosceles because it has two congruent sides.

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Solution:

Given: M is the midpoint of PK, PK ⊥ MB.

To Prove: △PKB is isosceles It is given that M is the midpoint of PK and PK ⊥ MB.

Proof: In △PKB and △KMB

1. Midpoints divide a segment into two congruent segments, so PM ≅ KM.

2. Since PK ⊥ MB and perpendicular lines intersect at right angles, ∠PMB and ∠KMB are right angles. Right angles are congruent, so ∠PMB ≅ ∠KMB.

3. The triangles share MB, and the reflexive property justifies that MB ≅ MB.

 Therefore, △PMB ≅ △KMB [by the SAS congruence theorem]

     BP ≅ BK  [C P C T]

4. △PKB is isosceles because it has two congruent sides.


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