Match the following items.If ∠ADB = 70°, then 1. m∠ABD     = 140 2. m (AB        = 20 3. m (AD        = 40

Accepted Solution

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°.