ABCD is a trapezium in which AB || CD and AD = BC (see Fig. 8.23). Show that
(i) ∠A = ∠B
(ii) ∠C = ∠D
(iii) ∆ABC ≅ ∆BAD
(iv) diagonal AC = diagonal BD
[Hint: Extend AB and draw a line through C parallel to DA intersecting AB produced at E.]

Solution:

Let us join BD and AC in the figure tas shown below.. ADCE is a parallelogram.

(i) AD = CE (Opposite sides of parallelogram AECD are equal)

However, AD = BC (Given)

Therefore, BC = CE

∠CEB = ∠CBE (Angles opposite to equal sides in a triangle are also equal)

Consider, parallel lines AD and CE where AE is the transversal.

∠BAD + ∠CEB = 180° [Co-Interior angles]

∠BAD + ∠CBE = 180° ... (1) [Since, ∠CEB = ∠CBE]

However, ∠ABC + ∠CBE = 180° (Linear pair angles) ... (2)

From Equations (1) and (2), we see that

∠BAD = ∠ABC

Thus, ∠A = ∠B

(ii) AB || CD

∠A + ∠D = 180° (Angles on the same side of the transversal)

Also, ∠C + ∠B = 180° (Angles on the same side of the transversal)

∴ ∠A + ∠D = ∠C + ∠B

However, ∠A = ∠B [Using the result obtained in (i)]

∴ ∠C = ∠D

(iii) In ∆ABC and ∆BAD,

AB = BA (Common side)

BC = AD (Given)

∠B = ∠A (Proved before)

∴ ∆ABC ≅ ∆BAD (SAS congruence rule)

(iv) Since ∆ABC ≅ ∆BAD,

∴ AC = BD (By CPCT)

☛ Check: NCERT Solutions Class 9 Maths Chapter 8

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Video Solution:

ABCD is a trapezium in which AB || CD and AD = BC (see Fig. 8.23). Show that (i) ∠A = ∠B (ii) ∠C = ∠D (iii) ∆ABC ≅ ∆BAD (iv) diagonal AC = diagonal BD [Hint: Extend AB and draw a line through C parallel to DA intersecting AB produced at E.]

NCERT Maths Solutions Class 9 Chapter 8 Exercise 8.1 Question 12

Summary:

If ABCD is a trapezium in which AB || CD and AD = BC, we have proved that ∠A = ∠B, ∠C = ∠D, △ABC ≅ △BAD by SAS congruence, and diagonal AC = diagonal BD.

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☛ Related Questions:

• ABCD is a rectangle in which diagonal AC bisects ∠A as well as ∠C. Show that:(i) ABCD is a square(ii) diagonal BD bisects ∠B as well as ∠D.
• In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see Fig. 8.20). Show that: (i) ΔAPD ≅ ΔCQB (ii) AP = CQ (iii) ΔAQB ≅ ΔCPD (iv) AQ = CP (v) APCQ is a parallelogram
• ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD (See the given figure). Show that i) ΔAPB ≅ ΔCQD ii) AP = CQ
• In ∆ ABC and ∆ 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 the given figure). 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) ∆ ABC ≅ ∆ DEF