AB=AC,∠ABD=60°,∠ADB=70°,∠BDC=40°,find∠DBC

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  • Thread starter Albert1
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In summary, the equation AB=AC means that the length of line segment AB is equal to the length of line segment AC. The angles ∠ABD, ∠ADB, and ∠BDC are related by intersecting at point D and forming a triangle. To find the value of ∠DBC, the angle sum property of triangles can be used. It is important to know the value of ∠DBC as it helps in understanding the properties and relationships of the given triangle. Other methods such as the law of sines or law of cosines can also be used to solve this problem, but basic geometry principles are enough to find the missing angle in this case.
  • #1
Albert1
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$Quadrilateral \,\,ABCD,\overline{AB}=\overline{AC} ,\angle ABD=60^o,\angle ADB=70^o,
\angle BDC=40^o,find \,\, \angle DBC=?$
 
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  • #2
Albert said:
$Quadrilateral \,\,ABCD,\overline{AB}=\overline{AC} ,\angle ABD=60^o,\angle ADB=70^o,
\angle BDC=40^o,find \,\, \angle DBC=?$
hint:
using the diagram:
construct a point E between $\overline{CD},and \,\, \angle ABE=70 ^o$
prove points C and E overlap
View attachment 6993
 

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  • #3
from previous diagram we have:
Points $A,B,E,D$ cocyclic, so $\overset{\frown}{AB}=\overset{\frown}{AE}=140^o$
we get $\overline{AB}=\overline{AE}=\overline{AC}$
and points C and E overlap (since points C,E,D colinear)
$\therefore \angle DBC=\angle DBE=10^o$
 

FAQ: AB=AC,∠ABD=60°,∠ADB=70°,∠BDC=40°,find∠DBC

1. What does the equation AB=AC mean in this context?

The equation AB=AC means that the length of the line segment AB is equal to the length of the line segment AC.

2. How are the angles ∠ABD, ∠ADB, and ∠BDC related to each other?

These angles are related by the fact that they all intersect at the point D and form a triangle. ∠ABD and ∠ADB are adjacent angles, while ∠ADB and ∠BDC are opposite angles.

3. How can you find the value of ∠DBC?

To find the value of ∠DBC, you can use the angle sum property of triangles, which states that the sum of all angles in a triangle is equal to 180 degrees. You can subtract the known angles ∠ADB and ∠BDC from 180 degrees to find the measure of ∠DBC.

4. Why is it important to know the value of ∠DBC?

The value of ∠DBC is important because it helps us understand the properties and relationships of the given triangle. It also allows us to accurately measure and calculate other angles and lengths within the triangle.

5. Are there any other methods to solve this problem?

Yes, there are other methods to solve this problem, such as using the law of sines or the law of cosines. These methods involve using trigonometric functions to find the missing angle or side length. However, the given information in this problem is sufficient to solve for ∠DBC using basic geometry principles.

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