Deducing Properties of Isosceles Triangles

Q.

In an isosceles triangle AB = AC and BA is produced to D, such that AB = AD. What is the value of $\angle BCD$?

1. ${45}^{\circ }$

2. ${60}^{\circ }$

3. ${90}^{\circ }$

4. ${70}^{\circ }$

Q.

Bisectors of the angles B and C of an isosceles triangle with AB = AC intersect each other at O. BO is produced to a point M. Then which of the following options is correct?

1. $\angle MOC=\angle BAC$
   

2. $\angle MOC=\angle ABC$
   

3. $\angle MOC=\angle MBC$
   

4. None of these

Q.

If in the following figure, AC = CD, AD = BD and $\angle C={58}^{\circ }$, then the measure of angle CAB is

1. ${90}^{\circ }$

2. ${91.5}^{\circ }$

3. ${92.5}^{\circ }$

4. ${93}^{\circ }$

Q.

In the given figure, AD = AB = AC, BD is parallel to CA and angle ACB = 65°. The measure of $\angle DAC$ is

Q.

If in the following figure, AC = CD, AD = BD and $\angle C={52}^{\circ }$, then the measure of angle DAB is

1. ${30}^{\circ }$

2. ${32}^{\circ }$

3. ${34}^{\circ }$

4. ${36}^{\circ }$

Q. Question 9
If $\Delta ABC\sim \Delta DFE,\angle A={30}^{\circ },\angle C={50}^{\circ },$ $AB=5cm,AC=8cm,andDF=7.5cm$ , then which of the following is true?

(A) $DE=12cm,\angle F={50}^{\circ }$
(B) $DE=12cm,\angle F={100}^{\circ }$
(C) $EF=12cm,\angle D={100}^{\circ }$
(D) $EF=12cm,\angle D={30}^{\circ }$

Q. Find the value of x in the following diagram. [4 MARKS]

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Q.

$△$ABC is an isosceles triangle, in which AB=AC. Side BA is produced to D such that AD=AB. $\angle$BCD is equal to ___.

1. 

2. 

3. Can not be determined

4. 

Q.

Bisectors of the angles B and C of an isosceles triangle with AB = AC intersect each other at O. BO is produced to a point M. Then which of the following options is correct?

1. $\angle MOC=\angle BAC$
   

2. $\angle MOC=\angle MBC$
   

3. $\angle MOC=\angle ABC$
   

4. $\angle MOC=\angle BCM$

Q.

Bisectors of the angles B and C of an isosceles triangle with AB = AC intersect each other at O. BO is produced to a point M. Which of the following is correct?

1. $\angle MOC=\angle BAC$
   

2. $\angle MOC=\angle ABC$
   

3. $\angle MOC=\angle MBC$
   

4. None of these

Q.

The corresponding sides of 2 similar triangles $△ABC$ and $△PQR$ are in the ratio 4:23. $\angle R$ is___________
Given that $\angle B$=${60}^{\circ }$ and $\angle A$= ${40}^{\circ }$.

1. 

2. 

3. 

4. 

Q.

In an isosceles triangle AB = AC and BA is produced to D, such that AB = AD. What is the value of $\angle BCD$?

1. ${90}^{\circ }$

2. ${60}^{\circ }$

3. ${70}^{\circ }$

4. ${45}^{\circ }$

Q.

An isosceles triangle ABC has AC = BC. CD bisects AB at D and $\angle CAB={55}^{\circ }$. Then $\angle DCB$ equals

Q.

In the given figure, AD = AB = AC, BD is parallel to CA and angle ACB = 65°. The measure of $\angle DAC$ is

1. ${50}^{\circ }$
2. ${80}^{\circ }$
3. ${130}^{\circ }$
4. ${150}^{\circ }$

Q.

Find the angle x from the figure if ABCD is a parallelogram in which $\angle A:\angle B=1:4$. Also, $\Delta$DEF is an isosceles triangle with DE = DF.

1. 36°

2. 54°

3. 18°

4. 44°

Q.

In an isosceles triangle ABC, if the measure of angle ACB is equal to that of angle CAB, then ______.

1. AB < BC

2. AB = AC

3. BC = AC

4. AB = BC

Q.

An isosceles triangle ABC has AC = BC. CD bisects AB at D and $\angle CAB={55}^{\circ }$. Then $\angle DCB$ equals

1. ${70}^{\circ }$
2. ${55}^{\circ }$
3. ${35}^{\circ }$
4. ${75}^{\circ }$

Q.

Find $x$ in the following figure:

1. ${73}^{\circ }$
2. ${42}^{\circ }$
3. ${54}^{\circ }$
4. ${63}^{\circ }$

Q. In the figure, given, $P$ is a point on $AB$ such that $AP:PB=4:3$. $PQ$ is parallel to $AC$.
(i) Calculate the ratio $PQ:AC$, giving reason for your answer
(ii) In triangle $ARC,\angle ARC={90}^o$ and in triangle $PQS,\angle PSQ={90}^o$. Given $QS=6cm$, calculate the length of $AR$.