Triangles ABC and CBD are not congruent because they have different side lengths and angles. However, BD does bisect angle ABC due to the isosceles property of triangle ADB.
No, triangle ABC is not congruent to triangle CBD. Although they share two sides (AB and CB), the third sides (AC and CD) are not equal, and the angles between these sides are also not congruent.
Different side lengths: AC and CD are not congruent. In the image, you can see that AC is the hypotenuse of right triangle ADC, while CD is a leg. Since the hypotenuse of a right triangle is always longer than its legs, AC must be greater than CD.
Different angles: The angles between AC and AB, and CD and CB, are not congruent. This is because these angles are opposite the unequal sides AC and CD. In right triangles, the angles opposite the longer side are always wider than the angles opposite the shorter side. Therefore, the angle at A in triangle ABC is wider than the angle at C in triangle CBD.
Therefore, based on the violation of both the side and angle congruence conditions, triangles ABC and CBD cannot be considered congruent.
As for the second part of your question, BD does bisect angle ABC. Here's why:
Triangle ADB is isosceles: Since AB and CB are congruent, triangle ADB is isosceles (a triangle with two equal sides).
Base angles of an isosceles triangle are congruent: In an isosceles triangle, the angles at the base (the angles opposite the two congruent sides) are always congruent. Therefore, angles ABD and BDC in triangle ABD are congruent.
BD bisects angle ABC: Since BD bisects angle BDC, it also bisects angle ABC, which is the sum of angles ABD and BDC.
Therefore, BD indeed bisects angle ABC.
The question probable may be:
Is triangle ABC congruent to triangle CBD? why or why not? does BD bisect angle abc give reasons?