Question

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12. ABCD is a trapezium in which AB || CD and AD=BC (see Fig. 8.23). Show that

(i) Angle A= angle B
(ii) Angle C= angle D
(iii) Triangle ABC congruent Triangle BAD
(iv) diagonal AC = diagonal BD

(Hint: Extend AB and draw a line through C parallel to DA intersecting AB produced at E.]
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Answer 1

Answer:

Answer

Given ABCD is trapezium where AD=BC.

(i) To prove: ∠A=∠B

we can see that AECD is a parallelogram, so sum of adjacent angles =180

o

→∠A+∠E=180

o

→∠A+x=180

o

→∠A=180

o

−x=∠B

Hence proved.

(ii) To prove: ∠C=∠D

sum of adjacent angles in parallelogram is π, so

→∠D∠C+180

o

−2x=180

o

→∠C+∠D=2x

Now

→∠B+∠C=180

o

→180

o

−x+∠C=180

o

=0 ∠C=x, so

∠D=x

And,

∠C=∠D

Hence proved.

(iii) ΔABC=ΔBAD

→ side AB is common.

→AD=BC (given)

so the angle including both the sides is also same,

∠A=∠B. So

ΔABC=ΔBAD (By SAS congruent Rule)

Hence proved.