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Given that ABCD is a parallelogram with diagonals intersecting at M. Segment PQ is drawn such that P lies on AB and Q lies on DC and it passes through M. Prove that PM is congruent to QM.​

Given That ABCD Is A Parallelogram With Diagonals Intersecting At M Segment PQ Is Drawn Such That P Lies On AB And Q Lies On DC And It Passes Through M Prove Th class=

Sagot :

Answer:

we thus have enough informnation to prove that the two triangles are congruent. Because of thism, one side of the triangles, Pm which is the congruent side to that of PQ must be equal as well and

PM = QM

Step-by-step explanation:

because it's a parrallelogram, Ad = BC and AB = DC.

P passes through both  AB and DC on a diagonal, angles created by P have oppoiste exterior angles of eachother.

the diagonals of AC create congruent angles at DMC and AMB and because P is cutting through them, the angles P cuts at DMQ and QMC are equal to that of AMP and PMB

that being said, we now see that the angles of APM triangle and AMC triangol are equal and because the diagonal of BD is is being cut by AC, which AD is parralel to BC  and AB to Dc, we know now that lines AM and MC are congruent

we thus have enough informnation to prove that the two triangles are congruent. Because of thism, one side of the triangles, Pm which is the congruent side to that of PQ must be equal as well and

thus PM = QM

The alternate interior angles formed by the parallel sides and a common

transversal are equal, such that ΔPBM and ΔQDM are congruent.

Correct response;

  • PM is congruent to QM  by CPCTC

How is the congruency of the segments determined?

The two column proof is presented as follows;

Statement [tex]{}[/tex]                                          Reason

ABCD is a parallelogram [tex]{}[/tex]                         Given

AC and DB are diagonals of ABCD [tex]{}[/tex]        Given[tex]{}[/tex]

M is the point of intersection of the diagonals [tex]{}[/tex] Given

MB ≅ DM, and AM ≅ CM;[tex]{}[/tex]  Properties of the diagonals of a parallelogram

∠PMB ≅ ∠DMQ [tex]{}[/tex]                                Vertical angles theorem

∠PBM ≅ ∠QDM [tex]{}[/tex]                                Alternate interior angles theorem

ΔPBM ≅ ΔQDM  [tex]{}[/tex]                               Angle-Side-Angle Congruency post.

PM ≅ QM [tex]{}[/tex]                                          CPCTC

ΔPBM is congruent to ΔQDM by Angle-Side-Angle congruency

postulate, given that they have two angles and an included side that are

congruent.

Therefore;

  • PM is congruent to QM by Corresponding Parts of Congruent Triangles are Congruent, CPCTC.

Learn more about congruency postulates here:

https://brainly.com/question/17003791

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