BD is the bisector of angle ABC . prove that the perpendicular distances from BD to AB and CB are equal.
triangles BDA and BDC are congruent.
To prove that the perpendicular distances from BD to AB and CB are equal, we can use the properties of angles and triangles. Here's how you can prove it:
Step 1: Given that BD is the bisector of angle ABC, we know that angle ABD is congruent to angle CBD.
Step 2: Draw a perpendicular line from BD to AB and label the point of intersection as E. Draw another perpendicular line from BD to CB and label the point of intersection as F.
Step 3: Consider triangles ABD and CBD. Since ABD is congruent to CBD (because BD is the bisector of angle ABC), we have two pairs of congruent angles, angle ABD ≅ angle CBD and angle ADB ≅ angle CDB.
Step 4: By the angle-angle congruence (AA) property, we can conclude that triangles ABD and CBD are similar.
Step 5: Since triangles ABD and CBD are similar, their corresponding sides are proportional. This means that the ratio of the perpendicular distance from BD to AB (DE) to the perpendicular distance from BD to CB (DF) is equal to the ratio of the corresponding sides AB and CB.
Step 6: Since the corresponding sides AB and CB are the same, the ratio of DE to DF is equal to 1.
Step 7: Therefore, the perpendicular distances from BD to AB and CB are equal, as DE ≅ DF.
By following these steps, you have proven that the perpendicular distances from BD to AB and CB are equal.