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Julie and Samuel were each asked to prove [tex]$m \angle CBD = 65^{\circ}$[/tex] with the given information. What type of proofs did they use?

Given:
[tex]\[
\begin{array}{l}
m \angle ABC = 90^{\circ} \\
m \angle ABD = 25^{\circ}
\end{array}
\][/tex]

Prove: [tex]$m \angle CBD = 65^{\circ}$[/tex]

\begin{tabular}{|l|l|}
\hline \multicolumn{2}{|c|}{ Julie's Proof } \\
\hline Statements & Reasons \\
\hline 1. [tex]$m \angle ABC = 90^{\circ}$[/tex] & 1. given \\
\hline 2. [tex][tex]$m \angle ABD = 25^{\circ}$[/tex][/tex] & 2. given \\
\hline 3. [tex]$m \angle ABD + m \angle CBD = 90^{\circ}$[/tex] & 3. angle addition postulate \\
\hline 4. [tex]$25^{\circ} + m \angle CBD = 90^{\circ}$[/tex] & 4. substitution property of equality \\
\hline 5. [tex][tex]$m \angle CBD = 65^{\circ}$[/tex][/tex] & 5. subtraction property of equality \\
\hline
\end{tabular}

Sagot :

GinnyAnsweridn
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].
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