ASA Criteria for Congruency of Triangles

Q.

If $a$ and $b$ be two unit vectors such that angle between them is $60°$. Then, $|a-b|$ is equal to?

1. $\surd 5$

2. $\surd 3$

3. $0$

4. $1$

Q.

If $a,b$ and $c$ be three unit vectors such that$a\times (b\times c)=1/2b$, $b$and $c$ being non - parallel. If ${\theta }_1$ is the angle between $a$ and $b$ and ${\theta }_2$ is the angle between $a$ and $c$, then

1. ${\theta }_1=\pi /6,{\theta }_2=\pi /3$

2. ${\theta }_1=\pi /3,{\theta }_2=\pi /6$

3. ${\theta }_1=\pi /2,{\theta }_2=\pi /3$

4. ${\theta }_1=\pi /3,{\theta }_2=\pi /2$

Q. Two triangles are formed on the same base and equal angles at the ends of the base. The triangles are congruent by _____ criterion.

1. AAA
2. ASA
3. SSS
4. RHS

Q. If $△ABC$ is an equilateral triangle. AO, BO and CO are angle bisectors, then ______.

1. AO = BO = CO
2. AO = AC
3. $△AOC\cong △COB\cong △BOA$
4. $△AOC≆△COB≆△BOA$

Q.

A right-angled triangle DEF is constructed with DE = 5 cm, ∠F = 90°, and DF = 4 cm. Choose the correct statement from the following.

1. DF is the hypotenuse of △DEF
2. EF = 3 cm
3. $\angle E+\angle D={170}^{\circ }$
4. EF = 2 cm

Q.

Question 147

If $\Delta PQR$ and $\Delta SQR$ are both isosceles triangle on a common base QR such that P and S lie on the same side of QR. Are $\Delta PSQ$ and $\Delta PSR$ congruent? Which condition do you use?

Q. If, PS is parallel to QR and PS = QR, then, which of the given pair of triangles are congruent?

1. $△$SRT and $△$RQT
2. $△$PST and $△$RQT
3. $△$PTS and $△$SRQ
4. $△$PST and $△$SRT

Q.
Line a is parallel to line b and line c is parallel to line d. Then,

1. QR = SP
2. PQ = RS
3. QR = PR
4. SP = RP

Q.

(a) DA bisects $\angle BAC$ and $\angle B=\angle C$. Prove that $\Delta BDA\cong \Delta CDA$.

(b) If these triangles are congruent, choose the property by which they are congruent.

[4 MARKS]

Q.

In the given figure, measures of sides of two triangles $ABC$ and $PQR$ are given. Examine whether the two triangles are congruent or not. If yes, write the congruence relation in symbolic form.

Q.

Question 98

State whether the statement is True or False.

If three angles of two triangles are equal, triangles are congruent.

Q.

$\angle ABC\cong \angle PQR.If\angle ABC=75°.Find\angle PQR$

Q.
What criteria can be used to prove the given triangles are congruent?

1. SAS
2. ASA
3. SSS
4. RHS

Q.

ABC is an isosceles triangle with $AB=AC$. Prove: [4 MARKS]

(i) $\Delta ADB\cong \Delta ADC$

(ii) $\angle BAD=\angle CAD$

(iii) $BD=CD$

Q. In the above figure, TQ = SR and $\angle$TQP = $\angle$SRQ.
If, $\angle$RQS = 55°, then, $\angle$QPT = ?.

1. 75°
2. 65°
3. 45°
4. 55°

Q.
Given above are two triangles $△$ABC and $△$DEF. Which of the following statements is correct?

1. $△$ABC $\cong △$DEF by SAS criteria
2. $△$ABC $\cong △$DEF by SSS criteria
3. $△$ABC $\cong △$DEF by ASA criteria
4. $△$ABC $\cong △$DEF by RHS criteria

Q. In the given figure, $\angle$DAO = $\angle$CBO and AO = BO. Identify the type of $△$DOC.

1. Scalene triangle
2. Right-angle triangle
3. Equilateral triangle
4. Isosceles triangle

Q.

In the given figure, if AB = BC and $\angle BAO=\angle BCO={90}^{\circ }$, then which of the following is true?

1. $△ABO\cong △CBO$ by RHS postulate
2. $△ABO\cong △CBO$ by ASA postulate
3. OA = OC
4. If $\angle ABO={60}^{\circ }$ then, $\angle CBO={60}^{\circ }.$

Q.

Hari was asked to bisect a given angle, $\angle$ BOA as shown. He has done the following steps:

He draws an arc with centre O and some radius such that it cuts OB and OA at D and C. Then he marks two more arcs with centres as C and D and radius more than $\frac{1}{2}CD$ as shown (intersecting in Y) but has no idea why.

Select the correct statement which explains him the reason.

Statement A : Because of equal radius, DY = CY and OD = OC. So $△$ DOY is congurent to $△$ COY

Statement B : Since OC = OD, $△$ ODC in an isosceles triangle, base angles are equal.

1. Statements A and B are true, only Statement A explains the reason.
2. Statements A and B are false.
3. Statements A and B are true, but none explains the reason.
4. Statements A and B are true, only Statement B explains the reason.

Q.

In fig. PQRS is a quadrilateral and T and U are respectively points on PS and RS such that
PQ=RQ,
∠PQT=∠RQU...(i)

∠TQS=∠UQS...(ii)
Prove that: QT=QU.

Q. Suppose ABC is an equiangular triangle.. Prove that it is equilateral. (You have seen earlier that an equilateral triangle is equiangular. Thus for triangles equiangularity is equivalent to equilaterality.)

Q. In the given figure, QTSR is a parallelogram. PQR is a straight line and QU = UT. Which of the following statements are correct?

1. $△PQU\cong △STU$
2. $QU=US$
3. $PQ=QR$
4. $PU=US$

Q. In the given figure, AC = CE and AB||ED. The value of x is .

1. 12
2. 15
3. 18
4. 20

Q. (a) In the given figure, show that $\Delta AMP\cong \Delta AMQ$.




(b)

In the given figure, AC = CE and AB $\parallel$ ED. The value of $x$ is ___ units.
[4 MARKS]

Q.

$△$STR is an isosceles triangle where ST=TR
Also PT = TR,
and $\angle$TSP = $\angle$TRQ
Find the angle of $△$PST that corresponds to $\angle$RQT of $△$RQT.

1. $\angle$PST
2. $\angle$TPS
3. $\angle$STP
4. Cannot be determined

Q. $OA=OB,OC=OD$, $\angle AOB=\angle COD$. Prove that $AC=BD$

Q. A right-angled triangle DEF is constructed with DE = 5 cm, ∠F = 90°, and DF = 4 cm. Choose the correct statement from the following.

1. DF is the hypotenuse of △DEF
2. $\angle E+\angle D={170}^{\circ }$
3. EF = 3 cm
4. EF = 2 cm

Q. In Fig, $10.18,PQRS$ is quadrilateral and $T$ and $U$ are respectively points on $PS$ and $RS$ such that $PQ=RQ,\angle PQT=\angle RQU$ and $\angle TQS=\angle UQS$. Prove that $QT=QU$.

Q. A right triangle DEF is constructed with DE = 5 cm, $\angle F={90}^{\circ }$ and DF = 4 cm. Choose the correct statement from the following.

1. DF is the hypotenuse of $\Delta$ DEF
2. $\angle E+\angle D={180}^{\circ }$
3. EF = 3 cm
4. $\angle E={90}^{\circ }$

Q. Triangle $PQR$ is right angled at $Q$ and has side lengths $PQ=14$ and $QR=48.$ If $M$ is the mid point of $PR$ and $\angle MQP=\theta ,$ then $\cos \theta$ is equal to:

1. $\frac{7}{50}$
2. $\frac{7}{25}$
3. $\frac{7}{24}$
4. $\frac{24}{25}$