Question

in the diagram below of triangle abc, d is the midpoint of \\(\overline{ac}\\) and e is the midpoint of \\(\overline{bc}\\). if m \\(\angle cab = 58 - 2x\\), and m\\(\angle cde = 2x + 54\\), what is the measure of \\(\angle cab\\)?

Explanation:

Step1: Identify Midline Theorem

Since D is the midpoint of \( \overline{AC} \) and E is the midpoint of \( \overline{BC} \), by the Midline Theorem (also known as the Midsegment Theorem), \( \overline{DE} \parallel \overline{AB} \). This means \( \angle CDE \) and \( \angle CAB \) are corresponding angles, so they are equal. Thus, \( m\angle CAB = m\angle CDE \).

Step2: Set Up Equation

Given \( m\angle CAB = 58 - 2x \) and \( m\angle CDE = 2x + 54 \), set them equal:
\( 58 - 2x = 2x + 54 \)

Step3: Solve for \( x \)

Add \( 2x \) to both sides:
\( 58 = 4x + 54 \)
Subtract 54 from both sides:
\( 4 = 4x \)
Divide by 4:
\( x = 1 \)

Step4: Find \( m\angle CAB \)

Substitute \( x = 1 \) into \( m\angle CAB = 58 - 2x \):
\( m\angle CAB = 58 - 2(1) = 58 - 2 = 56 \)

Answer:

\( 56 \)