In a triangle ABC,right angled at B, BD is drawn perpendicular to AC. prove that:
(i) angle ABD = angle c
(ii) angle CBD = angle A
To prove the given statements, we will use the properties of right-angled triangles and the geometric definition of angles.
(i) To prove that angle ABD is equal to angle C, we can use the property that if two angles of one triangle are equal to two angles of another triangle, then the corresponding third angles are also equal.
In triangle ABD and triangle BDC:
- Angle ABD is a right angle since BD is perpendicular to AC.
- Angle BDC is a right angle as it is part of the right-angled triangle ABC.
Since both triangles share a common angle (right angle at B) and have another pair of equal angles (angle ABD and angle BDC are vertically opposite angles), we can conclude that angle ABD is equal to angle BDC.
Additionally, in triangle ABC, angle A + angle B + angle C = 180 degrees (sum of interior angles of a triangle). Since we know that angle B is 90 degrees (right angle), we can substitute it into the equation:
angle A + 90 degrees + angle C = 180 degrees
angle A + angle C = 90 degrees.
Since angle ABD is equal to angle BDC, we can substitute angle ABD with angle BDC in the equation:
angle BDC + angle C = 90 degrees.
Now, replace angle BDC in the equation with angle ABD:
angle ABD + angle C = 90 degrees
Therefore, angle ABD is equal to angle C.
(ii) To prove that angle CBD is equal to angle A, we can use the same logic as above.
In triangle ABD and triangle BDC:
- Angle ABD is a right angle since BD is perpendicular to AC.
- Angle BDC is a right angle as it is part of the right-angled triangle ABC.
Since both triangles share a common angle (right angle at B) and have another pair of equal angles (angle ABD and angle BDC are vertically opposite angles), we can conclude that angle ABD is equal to angle BDC.
Now, consider triangle ABC. By the sum of interior angles of a triangle, angle A + angle B + angle C = 180 degrees. Since angle B is 90 degrees (right angle), we can substitute it into the equation:
angle A + 90 degrees + angle C = 180 degrees
angle A + angle C = 90 degrees.
We have already established that angle ABD is equal to angle BDC. Therefore, we can substitute angle ABD with angle BDC in the equation:
angle BDC + angle C = 90 degrees
Now, replace angle BDC in the equation with angle ABD:
angle ABD + angle C = 90 degrees
Since angle ABD is a right angle, it measures 90 degrees. Therefore, we can rewrite the equation:
90 degrees + angle C = 90 degrees
Subtract 90 degrees from both sides of the equation:
angle C = 0 degrees
Therefore, angle C is equal to 0 degrees.
In conclusion, we have proved both statements:
(i) angle ABD = angle C
(ii) angle CBD = angle A