Question

HELP GEOMETRY PLZ!!!

Complete the following proof.


Given: QK=16, PK=26, MK=65, KL=40

Prove: ∆QPK~∆LMK

Answer 1

Statement 2.: QK/KL=16/40; PK/MK=26/65

Statement 3.: QK/KL=2/5=PK/MK

Reason 4: Vertical angles

Reason 5: SAS

Explanation:

Statement 2:

The smaller of the sides adjacent to <PKQ (QK=16) in triangle PKQ is corresponding with the smaller of the sides adjacent to <MKL (KL=40) in triangle MKL.

The larger of the sides adjacent to <PKQ (PK=26) in triangle PKQ is corresponding with the larger of the sides adjacent to <MKL (MK=65) in triangle MKL.

Ratio of the corresponding sides:

`(QK)/(KL)=(16)/(40)`

`(PK)/(MK)=(26)/(65)`

Statement 3. Simplifying ratios:

Dividing the numerator and denominator by 8:

`(QK)/(KL)=((16)/(8))/((40)/(8))\\ (QK)/(KL)=(2)/(5)`

Dividing the numerator and denominator by 13:

`(PK)/(MK)=((26)/(13))/((65)/(13))\\ (PK)/(MK)=(2)/(5)`

Reason 4: <MKL is conguent with <PKQ because they are vertical angles (angles opposite by the vertex)

Reason 5: The two triangles (∆QPK y ∆LMK) have a congruent angle (<MKL with <PKQ) and the corresponding sides including this angle proportionals (QK/KL=2/5=PK/MK), then the two triangles (∆QPK y ∆LMK) are similar. This case is SAS.