We are given that $m\angle ABC = 77^\circ$, and we are tasked with finding $m\angle ABD$ and $m\angle CBD$. To solve this, we need additional details about the relationship between points $A$, $B$, $C$, and $D$, such as whether $D$ is a point on $\overrightarrow{BC}$, or if there is some angle-splitting taking place.

Typically, in these types of problems, $\angle ABC$ is split into $\angle ABD$ and $\angle CBD$, and we assume that:

$m\angle ABC = m\angle ABD + m\angle CBD$

But without further information, we can't split $\angle ABC$ into $\angle ABD$ and $\angle CBD$ in a specific ratio or amount.

Could you provide additional details, such as the position of point $D$ or how the angles are related?

5 Related Questions:

1. If $\angle ABD = \angle CBD$, what would their measures be?
2. How would the result change if $D$ is on the angle bisector of $\angle ABC$?
3. What if $D$ is outside triangle $ABC$? How would that affect the angle measures?
4. Can we apply any known theorems (e.g., Angle Bisector Theorem) to this problem?
5. What happens to the angles if a triangle is formed with points $A$, $B$, and $C$?

Tip:

Always try to visualize geometric problems with a diagram to better understand the relationships between the points and angles.