Problems on Congruent Triangles - Test Questions

Question 11

Question:   In the figure BA=BC, prove that AD = CE.

Answer:    Given:
BA = BC, CE ^ AB, AD ^ BC
To prove:
AD = CE
Proof:
In triangles AEC and ADC,
( Given)
AC=AC ( common side)
(AB=BC)
(given)

Question 12

Question:   In the adjoining figure, if BAC and DEF are right angles triangles, prove that AB = EF.

Answer:    Given:
In triangles ABC and DEF,
AC=DE

To prove:
AB=EF
Proof:
In triangles DEF and ABC,

AC=DE ( given)
( given)
(ASA Postulate)

Hence, AB = EF (c.p.c.t)

Question 13

Question:   Select congruent triangles in the following set of triangles and state the congruency condition in each case.

Answer:    Case I:

In D III right angle is not formed between 3 and 4.
They are not congruent triangles
Case II:

In D I side 3 is not the included side.
DI is not congruent to DII and DIII.

Question 14

Question:   ABCD is a quadrilateral in which DF and BE are perpendicular drawn on the diagonal AC. If AE=FC and BC=AD, prove that BE=FD.

Answer:    Given:
ABCD is a quadrilateral, diagonal AC is drawn
BE ^ AC and DF ^ AC.
AE=FC and BC=AD.
To prove:
BE=FD
Proof:
Compare triangles BEC and AFD,
Hyp. BC= hyp. AD (given)

AE=FC ( given)
AC-AE=AC-FC
i.e., EC=AF

Question 15

Question:   In the adjoining figure, O is the mid-point of AB and CD. Prove that AC=BD and AC||BD.

Answer:    Given:

AC and BD are joined.
To prove:
AC=BD
Proof:
In triangles AOC and BOD,

Question 16

Question:   In the adjoining figure, AB and CD intersect at O such that AO=OD and OB=OC. Prove that AC=BD.

Answer:    Given:
AB and CD intersect at O such that AO=OD and OB=OC. AC and BD are joined.
To prove:

Proof:
In triangles AOC and BOD,





AC=BD

Question 17

Question:   In the adjoining figures, two sides AB and BC and the median AD of D ABC are equal respectively to the two sides PQ and

Answer:    Given:
In triangles ABC and PQR,



To prove:

Proof:
In triangles ABD and PQM
AB=PQ ( given)
AD=PM ( given)
BD=QM
(BC=QR and AD and PM are medians of BC and QR)










(When equals are subtracted from equals the retaining parts are equal)
Now compare triangles ADC and PMR,
(given)
DC = MR (BC=QR)
(given)
( Proved above)

Question 18

Question:   In the adjoining figure, AB=AC, BE and CF are respectively,

Answer:    Given:

To prove:

Proof:
In triangles EBC and FCB,
BC=BC ( common side)
(AB=AC given)
( base angles are equal)

Question 19

Question:   In the adjoining figures, AB=AC and DB=DC. Prove that .

Answer:    Given:
With reference to both fig(a) and (b)

To prove:

Construction:
Join AD.
Proof:
Comparing triangles ABD and ADC,

Question 20

Question:   D ABC is an isosceles triangle with AB=AC side BA is produced

Answer:    Given:
In D ABC, AB=AC, DA is produced such that AB=AD.
To prove:

Proof:
In D ABC, AB = AC ( given)
( base angles of an isosceles triangle are equal)





In D ADC,

AC=AD (given)

( base angles of an isosceles triangle)

(angle sum property)








Adding (i) and (ii),


= 180 - 90
= 90^o