Q. 104.4( 335 Votes )

# ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD (see Fig. 8.21). Show that

(i) Δ APB ≅Δ CQD

(ii) AP = CQ

Answer :

(i) In ΔAPB and ΔCQD,

∠APB = ∠CQD (Each 90°)

AB = CD (Opposite sides of parallelogram ABCD)

∠ABP= ∠CDQ (Alternate interior angles for AB || CD)

ΔAPB ΔCQD (By AAS congruency)

(ii) By using the above result

ΔAPB ΔCQD, we obtain

CPCT:**Corresponding**parts of

**congruent**triangles are

**congruent**

AP = CQ (By CPCT)

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PREVIOUSIn parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see Fig. 8.20). Show that:(i) Δ APD ≅Δ CQB(ii) AP = CQ(iii) Δ AQB ≅Δ CPD(iv) AQ = CP(v) APCQ is a parallelogramNEXTIn Δ ABC and Δ DEF, AB = DE, AB || DE, BC = EF and BC || EF. Vertices A, B and C are joined to vertices D, E and F respectively (see Fig. 8.22). Show that(i) Quadrilateral ABED is a parallelogram(ii) Quadrilateral BEFC is a parallelogram(iii) AD || CF and AD = CF(iv) Quadrilateral ACFD is a parallelogram(v) AC = DF(vi) Δ ABC ≅Δ DEF.

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