To solve for [tex]\( m \angle CBD \)[/tex] given the information [tex]\( m \angle ABC = 90^\circ \)[/tex] and [tex]\( m \angle ABD = 25^\circ \)[/tex], follow these steps:
1. Given: [tex]\( m \angle ABC = 90^\circ \)[/tex] and [tex]\( m \angle ABD = 25^\circ \)[/tex].
2. Angle Addition Postulate: This postulate states that if a point D lies in the interior of [tex]\( \angle ABC \)[/tex], then
[tex]\[
m \angle ABD + m \angle CBD = m \angle ABC
\][/tex]
Applying this postulate, we have:
[tex]\[
m \angle ABD + m \angle CBD = m \angle ABC
\][/tex]
3. Substitution Property of Equality: Substitute the given values into the equation:
[tex]\[
25^\circ + m \angle CBD = 90^\circ
\][/tex]
4. Solving for [tex]\( m \angle CBD \)[/tex]: To find [tex]\( m \angle CBD \)[/tex], isolate [tex]\( m \angle CBD \)[/tex] on one side of the equation:
[tex]\[
m \angle CBD = 90^\circ - 25^\circ
\][/tex]
5. Conclusion:
[tex]\[
m \angle CBD = 65^\circ
\][/tex]
Thus, the measure of [tex]\( \angle CBD \)[/tex] is [tex]\( 65^\circ \)[/tex].