Question

in the diagram below of triangle bcd, e is the midpoint of \\(\overline{bd}\\) and f is the midpoint of \\(\overline{cd}\\). if m\\(\angle dcb = 87 + 3x\\), and m\\(\angle dfe = -8x + 164\\), what is the measure of \\(\angle dcb\\)?

Explanation:

Step1: Identify Midsegment Theorem

Since E is the midpoint of \( \overline{BD} \) and F is the midpoint of \( \overline{CD} \), by the Midsegment Theorem, \( \overline{EF} \parallel \overline{BC} \). Thus, \( \angle DFE \) and \( \angle DCB \) are corresponding angles, so they are equal. So we set \( 87 + 3x = -8x + 164 \).

Step2: Solve for x

Combine like terms: \( 3x + 8x = 164 - 87 \).
Simplify: \( 11x = 77 \).
Divide both sides by 11: \( x = \frac{77}{11} = 7 \).

Step3: Find \( m\angle DCB \)

Substitute \( x = 7 \) into \( 87 + 3x \):
\( 87 + 3(7) = 87 + 21 = 108 \).

Answer:

\( 108 \)