Construction of an Angular Bisector and its Proof

Q.
Ashu has constructed the bisector of AB in such a way that AO=BO and AP=BP. The value of $\angle AQO$ is.

1. ${90}^{\circ }$
2. ${75}^{\circ }$
3. ${82.5}^{\circ }$
4. ${97.5}^{\circ }$

Q. The bisector of an angle lies

1. Any where in the plane
2. On the arms of the angle
3. To the exterior of the angle
4. To the interior of the angle

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 respectively. Then he draws two more arcs with centres as C and D and radius more than $\frac{1}{2}$CD as shown (intersecting at Y) but has no idea why.

Select the correct statement which explains to him the reason for his construction.

Statement A : Because of equal radius, DY = CY and OD = OC. So $△$ DOY is congruent 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 true, only Statement B explains the reason.
3. Statements A and B are true, but none explains the reason.
4. Statements A and B are false.

Q. Which congruency rule is used to prove the construction of an angular bisector.

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

Q. Draw an angle of ${80}^{\circ }$ with the help of protractor. Then, which of the following angles can be constructed using ${80}^{\circ }$ ?

1. 40°
2. 160°
3. 110°
4. 150°

Q. How many angler bisectors are drawn to construct an angle of ${22.5}^{\circ }$

1. 1
2. 2
3. 3
4. 4

Q. Observe the given construction, AF is the angular bisector of $\angle BAE$ and AC is the angular bisector of $\angle BAF$. If $\angle BAE$ = ${60}^{\circ }$, then $\angle BAC$ will be

1. ${60}^{\circ }$
2. ${15}^{\circ }$
3. ${10}^{\circ }$
4. ${30}^{\circ }$

Q. In the given figure, OE = OD. Which of the following are true?

1. OX is angle bisector $\angle AOB$
2. C is the midpoint of OX
3. $\angle AOX=\angle AOB$
4. $2\angle AOX=\angle AOB$

Q. In which of the following situations an angle bisector cannot be constructed to bisect the angle formed between two given lines?

1. Two parallel lines
2. Two perpendicular lines
3. None of the above
4. Two intersecting lines

Q.

Your aim is to construct an angle $\angle$ YPX = ${30}^{\circ }$ using a compass and ruler only. What is the total number of arcs to be drawn to construct an angle of ${30}^{\circ }$ after drawing the base ray PX?

1. 3 arcs
2. 4 arcs
3. 6 arcs
4. 5 arcs