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.
Answers
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 CPCTCHow 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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