SSS Criteria for Congruency

Q. In right triangle ABC, right angled at C, M is the mid-point of hypotenuse AB. C is joined to M and produced to a point D such that DM = CM. Point D is joined to point B (see the given figure). Show that:
$CM=\frac{1}{2}AB$

Q.

AB is a line segement. P and Q are points on opposite sides of AB such that each of them is equidistant from the points A and B (See Fig. 12.26). Show that the line PQ is perpendicular bisector of AB.

Q.

$\Delta ABCand\Delta DBC\text{are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC. If AD is extended to intersect BC at E, show that}\left(i\right)\Delta ABD\cong \Delta ACD\left(ii\right)\Delta ABE\cong \Delta ACE\left(iii\right)\text{AE bisects}\angle A\text{as well as}\angle D\left(iv\right)\text{AE is the perpendicular bisector of BC.}$

Q.

The areas of two similar triangles $ABC$and$DEF$ are $144c{m}^2$ and $81c{m}^2,$respectively. If the longest side of larger triangle $ABC$ is $36cm$, then the longest side of the smaller triangle $DEF$ is

1. $20cm$

2. $26cm$

3. $27cm$

4. $30cm$

Q. If three sides of one triangle are equal to the three sides of another triangle, then, the two triangles are congruent by SSS congruence condition.

1. True
2. False

Q.

Choose the correct answer in each of the following:

$\text{If AB = QR, BC = RP and CA = PQ then which of the following holds?}\left(a\right)\Delta ABC\cong \Delta PQR\left(b\right)\Delta CBA\cong \Delta PQR\left(c\right)\Delta CAB\cong \Delta PQR\left(d\right)\Delta BCA\cong \Delta PQR$

Q.

The diagonal of a rectangle divides it into 2 congruent triangles.

1. True

2. False

Q.

Which of the following is not a sufficient condition for two triangles to be congruent.

1. Corresponding sides are equal

2. Corresponding angles are equal

3. Two sides and included angles are respectively equal

4. Two angles and included sides are respectively equal

Q.

O is the centre of the circle. If $\angle$BAC= ${50}^{\circ }$, find ∠OBC.

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Q.

In the figure, it is given that AB = CD and AD = BC. Prove that $\Delta ADC\cong \Delta CBA.$

Q.

The given figure shows a circle with centre O. P is mid-point of chord AB.

Show that OP is perpendicular to AB.

Q.

$\text{In the given figure, AB = AC and OB =OC. Then,}\angle ABO:\angle ACO=?\left(a\right)1:1\left(b\right)2:1\left(c\right)1:2\left(d\right)\text{none of these}$

Q.

$\text{In the given figure, ABCD is a square and P is a point inside it such that PB = PD. Prove tht CPA is a straight line.}$

Q.

If the areas of two similar triangles are in the ratio $25:64$, write the ratio of their corresponding sides.

Q.

In a $\Delta PQR,ifPQ=QRandL,M$ and N are the mid-points of the sides PQ, QR and RP respectively. Prove that LN = MN.

Q.

In $\Delta$ ABC, AB = AC and the bisectors of angles B and C intersect at point O. Prove that :

(i) BO = CO

(ii) AO bisects $\angle BAC$

Q. P is a point equidistant from two lines l and m intersecting at point A. Show that the line AP bisects the angle between them.

Q.

Given that ACBD is a kite. By which congruency property are the triangles ACB and ADB congruent?

1. SSS property

2. RHS property

3. ASA property

4. SAS property

Q. Question 7
If $\Delta PQR\cong \Delta EDF$, then is it true to say that PR = EF ? Give reason for your answer.

Q.

$In\Delta \text{ABC, AB = AC and the bisectors of}\angle Band\angle C\text{meet at a point O. Prove that BO = CO and the ray AO is the bisector of}\angle A.$

Q. If 3 sides of one triangle are equal to the three sides of another triangle, then the two triangles are .

1. congruent by SSS
2. congruent by SAA
3. congruent by SAS
4. congruent by SSA

Q.

In two triangles ABC and ADC, if AB = AD and BC = CD. are they congruent?

Q.

If $△ABC\cong △DEF,$then write their corresponding parts which are congruent.

Q. Question1
Which of the following is not a ciriterion for congruence of triangles?
(A) SAS
(B) ASA
(C) SSA
(D) SSS

Q. If $\Delta \text{XYZ}$ and $\Delta \text{ABC}$ are congruent, then which of the following is true?

1. XY = AB, YZ = BC and ZX = AC
2. Both the triangles can overlap
3. Area($\Delta \text{ABC})$ = Area($\Delta \text{XYZ})$
4. All of the above.

Q.

In the given figure, AB = AC. Prove that :

(i) DP= DQ

(ii) AP = AQ

(iii) AD bisects angle A

Q.

By which of the following criterion two triangles cannot be proved congruent ?

1. AAA

2. SSS

3. SAS

4. ASA

Q. What is an axiom?

Q. Question 5
Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.

Q. In triangleABC, AB is equal to AC , and the bisector of angle B and C intersect at point O prove that BO=CO and the ray AO is the bisector of angleBAC