Congruency of Triangles

Q.

In the given figure, ABCD is a quadrilateral in which AB = AD and BC = DC. Prove that (i) AC bisects $\angle Aand\angle C$, (ii) BE = DE, (iii) $\angle ABC=\angle ADC$.

Q.

In the adjoining figure, $ABCD$ is a square and $\Delta EDC$ is an equilateral triangle. Prove that (i) $AE=BE,$ (ii) $\angle DAE={15}^{\circ }$.

Q. Corresponding sides of congruent triangles in short we write c.s.c.t. and corresponding angles of congruent triangles in short we write ______.

1. angles
2. sides
3. (a) and (b) above.
4. None of the above

Q.

In the adjoining figure, ABCD is a parallelogram in which AB is produced to E so that BE = AB. Prove that ED bisects BC.

Q. If the length of three medians of a triangle are equal, Let us prove that the triangle is an isosceles triangle.

Q. The side length of 2 cm, 3 cm, and 4 cm can be the sides of

1. Scalene triangle
2. Isosceles triangle
3. Equilateral triangle
4. None of the above

Q. Abcd is a parallelogram , AD is produced to E so that DE =DC and EC produced meets AB produced in F prove that BF =BC

Q.
Points P and Q have been taken on opposite sides AB and CD respectively of a parallelogram ABCD such that AP = CQ. Which of the following is correct?

1. AC = CD
2. AC and PQ bisect each other.
3. AC = PQ
4. None of the above

Q.

In the adjoining figure, ABCD is a parallelogram and E is the midpoint of side BC. If DE and AB when produced meet at F, prove that AF = 2AB.

Q. Any two triangles are said to be congruent if they have __ .

1. opposite sides equal
2. equal angles
3. same measures
4. equal sides

Q. 43. In triangle ABC, angle BAC=90 degree and AB=BC.seg AP is perpendicular on side BC. B-P-C.D is any point on side BC. prove that 2AD2=BD2+CD2

Q. In the given figure, $\square$ABCD is a parallelogram. Point E is on the ray AB such that BE = AB then prove that line ED bisects seg BC at point F.

Q.

$P$ and $Q$ are the mid-points of the opposite sides $AB$ and $CD$ of a parallelogram $ABCD.AQ$intersects $DP$ at $S$ and $BQ$ intersects $CP$ at $R$. Show that $PRQS$ is a parallelogram.

Q.

In a triangle, halves of two equal side is are equal or not ?

Q. Two triangles are congruent to each other as given in the figure by RHS criteria, then choose the correct option(s).

1. BC $\ne$ DE
2. BC = DE
3. DE = 4 cm
4. $\angle$ B $\ne$ $\angle$ E

Q.

In the adjoining figure, ABCD and PQRC are rectangles, where Q is the midpoint of AC. Then DP is equal to

1. DP > PC

2. DP = PC

3. DP = ¹/₃ DC

4. DP < PC

Q. In figure ABC is an isosceles triangle in which AB = AC. CP || AB and AP is the bisector of exterior $\angle CADof△ABC$. Prove that $\angle PAC=\angle BCA$ and ABCP is a parallelogram.

Q.

Question 64

Observe all the four triangles FAB, EAB, DAB and CAB as shown in the figure.

State whether the statement is True or False.

All triangles are equal in area.

Q.

In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see the given figure).

Show that:

(i) ΔAPD ≅ ΔCQB

(ii) AP = CQ

(iii) ΔAQB ≅ ΔCPD

(iv) AQ = CP

(v) APCQ is a parallelogram

Q.

AD is an altitude of an isosceles triangles ABC in which AB = AC. Show that

(i) AD bisects BC (ii) AD bisects ∠A.

Q. ABCD is a parallelogram, AD is produced to E so that DE = DC and EC produced meets AB produced in F. Prove that BF = BC

Q. In ΔPQR $\cong$ ΔEFD then ED =

(a) PQ

(b) QR

(c) PR

(d) None of these

Q. in a triangle abc d e f are midpoints of the sides bc, ca, ab and p is a point on bc such that ap is perpendicular to bc if angle def = 50 then angle fpd=

Q.

In the given figure, ABCD is a quadrilateral in which AB = AD and BC = DC. Prove that (i) AC bisects $\angle Aand\angle C$, (ii) BE = DE, (iii) $\angle ABC=\angle ADC$.

Q. In figure, ABCD is a parallelogram, P and Q are midpoints of side AB and DC respectively, then prove APCQ is a parallelogram

Q. E and F are mid-points of sides AB and CD, respectively of a parallelogram ABCD. AF and CE intersect diagonal BD in P and Q, respectively. Prove that diagonal BD is trisected at P and Q.

Q. $BE$ and $CF$ are two equal altitudes of a triangle $ABC$. Using RHS congruence rule, prove that the triangle $ABC$ is isosceles.

Q. If the diagonals of a quadrilateral bisect each other, prove that the quadrilateral is a parallelogram.

Q. Two triangles are congruent to each other as given in the figure by RHS criteria, then find the values of BC and DE. [3 Marks]

Q. 3. ABCD and PQRC are rectangles and Q is the mid point of AC. Show that P is the mid point of DC and R is the mid point of BC .also find ratio of at(ABCD)and at( PQRC)