Answer:
1. ∠ABD = 20°.
2. Arc AB = 140°.
3. Arc AD = 40°.
Step-by-step explanation:
Given information: ∠ADB = 70°. BD is diameter.
According to Central angle theorem, the central angle from two chosen points A and B on the circle is always twice the inscribed angle from those two points.
By Central angle theorem,
[tex]\angle DAB=90^{\circ}[/tex]
Using angle sum of property in triangle ADB we get,
[tex]\angle ADB+\angle DAB+\angle ABD=180^{\circ}[/tex]
[tex]70^{\circ}+90^{\circ}+\angle ABD=180^{\circ}[/tex]
[tex]\angle ABD=20^{\circ}[/tex].
Draw a line segment AO.
In triangle AOD, AO=OD, so
[tex]\angle ODB=\angle OAD=70^{\circ}[/tex]
Using angle sum property in triangle AOD,
[tex]\angle AOD+\angle ODA+\angle OAD=180^{\circ}[/tex]
[tex]\angle AOD+70^{\circ}+70^{\circ}=180^{\circ}[/tex]
[tex]\angle AOD=40^{\circ}[/tex]
Therefore length of arc AD is 40°.
The angle AOD and AOB are supplementary angles.
[tex]\angle AOD+\angle AOB=180^{\circ}[/tex]
[tex]40^{\circ}+\angle AOB=180^{\circ}[/tex]
[tex]\angle AOB=140^{\circ}[/tex]
Therefore length of arc AB is 140°.
