Which of the following pairs of triangles are congruent? In each case, state the
condition of congruency:
(a) In 𝚫 ABC and 𝚫 DEF, AB=DE, BC=EF and ∠B=∠E.
(b) In 𝚫 ABC and 𝚫 DEF, ∠B=∠E = 90° ; AC=DF and BC=EF.
(c) In 𝚫 ABC and 𝚫 QRP, AB=QR, ∠B=∠R and ∠C=∠P.
(d) In 𝚫 ABC and 𝚫 PQR, AB=PQ, AC=PR, and BC=QR.
(e) In 𝚫 ABC and 𝚫 PQR, BC=QR, ∠A=90°, ∠C=∠R=40o and ∠Q=50°.
In a triangle ABC, D is mid-point of BC; AD is produced up to E so that DE =
AD. Prove that:
(i) 𝚫 ABD and 𝚫 ECD are congruent.
(iii) AB is parallel to EC.
A triangle ABC has ∠B=∠C.
(i) The perpendiculars from the mid-point of BC to AB and AC are equal.
(ii) The perpendiculars form B and C to the opposite sides are equal.
A line segment AB is bisected at point P and through point P another line
segment PQ, which is perpendicular to AB, is drawn. Show that: QA=QB.
If AP bisects angle BAC and M is any point on AP, prove that the
perpendiculars drawn from M to Ab and AC are equal.
From the given diagram, in which ABCD is a parallelogram, ABL is a line
segment and E is mid-point of BC.
In the following figure, AB=AC and AD is perpendicular to BC. BE bisects angle
B and EF is perpendicular to AB.
In the figure, given below, triangle ABC is a right-angled triangle at B. ABPQ and
ACRS are squares. Prove that:
(i) 𝚫ACQ and 𝚫ASB are congruent.
In a 𝚫 ABC, BD is the median to the side AC, BD is produced to E such that
Prove that: AE is parallel to BC.
In the adjoining figure, QX and RX are the bisectors of the angles Q and R
respectively of the triangle PQR.
If XS⊥QR and XT⊥PQ; prove that:
In parallelogram ABCD, the angles A and C are obtuse. Points X and Y are
taken on the diagonal BD such that the angles XAD and YCB are right angles.
Prove that: XA=YC.
ABCD is a parallelogram. The sides AB and AD are produced to E and F
respectively, such that AB=BE and AD=DF.
Prove that: 𝚫BEC≅ 𝚫DCF
In the following figures, the sides AB and BC and the median AD of triangle
ABC are respectively equal to the sides PQ and QR and the median PS of the
triangle PQR. Prove that 𝚫ABC and 𝚫PQR are congruent.
The following figure shows a triangle ABC in which AB=AC. M is a point on AB
and N is a point on AC such that BM=CN.
(ii) 𝚫AMC ≅ 𝚫ANB
(iv) 𝚫BMC ≅ 𝚫CNB
In a triangle, ABC, AB=BC, AD is perpendicular to side BC and CE is
perpendicular to side AB. Prove that:
PQRS is a parallelogram. L and M are points on PQ and SR respectively such
that PL=MR. Show that LM and QS bisect each other.
In the following figure, ABC is an equilateral triangle in which P is parallel to
AC. Side AC is produced up to point R so that CR=BP. Prove that QR bisects
AD and BC are equal perpendiculars to a line segment AB. If AD and BC are
on different sides of AB prove that CD bisects AB.
In 𝚫ABC, AB=AC and the bisectors of angles B and C intersect at point O.
(ii) AO bisects angle BAC.
A point O is taken inside a rhombus ABCD such that its distance from the
vertices B and D are equal. Show that AOC is a straight line.
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