To find \(m\angle ABD\), we need to add the measures of \(\angle ABC\) and \(\angle CBD\):
\[m\angle ABD = m\angle ABC + m\angle CBD\]
Given:
\[m\angle ABC = 30^ \circ \]
\[m\angle CBD = 45^ \circ \]
So,
\[m\angle ABD = 30^ \circ + 45^ \circ = 75^ \circ \]
Therefore, \(m\angle ABD = 75^ \circ \).

Supplemental Knowledge

In geometry, the measure of an angle formed by two adjacent angles can be found by adding the measures of those two angles. This principle is known as the Angle Addition Postulate.

1. Angle Addition Postulate:
   • If point \(C\) lies in the interior of \(\angle ABD\), then \(m\angle ABC + m\angle CBD = m\angle ABD\).
2. Application:
   • Given two adjacent angles, you can find the measure of the larger angle by summing the measures of the smaller angles.

Applied Knowledge

Learning how to add angles can be essential in many fields - architecture, engineering and even everyday tasks like measuring rooms for furniture placement or finding directions when traveling.

Mastering geometric principles like angle addition can significantly enhance your problem-solving skills and spatial reasoning. At UpStudy, we offer resources that help you practice these skills through various exercises and personalized feedback. For detailed guidance on improving your understanding of geometric concepts, explore UpStudy’s live tutor question bank or our AI-powered problem-solving services.
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