Question

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 = -8x + 93\\), and m\\(\angle cde = 99 - 9x\\), what is the measure of \\(\angle cde\\)?
answer attempt 1 out of 2
m\\(\angle cde = \square^\circ\\)
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Explanation:

Step1: Identify the relationship (Midline Theorem)

Since \( D \) is the midpoint of \( AC \) and \( E \) is the midpoint of \( BC \), by the Midline Theorem, \( DE \parallel AB \). Therefore, \( \angle CDE \) and \( \angle CAB \) are supplementary (same - side interior angles). So, \( m\angle CDE + m\angle CAB=180^{\circ} \).

Step2: Set up the equation

We know that \( m\angle CAB = - 8x + 93 \) and \( m\angle CDE=99 - 9x \). Substitute these into the supplementary angle equation:

$$ (-8x + 93)+(99 - 9x)=180 $$

Step3: Simplify and solve for \( x \)

Combine like terms:

$$ -8x-9x + 93 + 99=180 $$

$$ -17x+192 = 180 $$

Subtract 192 from both sides:

$$ -17x=180 - 192 $$

$$ -17x=-12 $$

Divide both sides by - 17:

$$ x=\frac{12}{17} $$

Wait, this seems wrong. Wait, if \( DE\parallel AB \), then \( \angle CDE \) and \( \angle CAB \) should be equal? Wait, no, let's re - examine the diagram. If \( D \) is the midpoint of \( AC \) and \( E \) is the midpoint of \( BC \), then \( DE\parallel AB \), so \( \angle CDE=\angle CAB \) (corresponding angles). Oh! I made a mistake in the angle relationship.

Let's correct Step1: Since \( DE\parallel AB \), \( \angle CDE \) and \( \angle CAB \) are corresponding angles, so \( m\angle CDE=m\angle CAB \).

Step1 (corrected): Identify the correct relationship

By the Midline Theorem, \( DE\parallel AB \). So, \( \angle CDE\) and \( \angle CAB \) are corresponding angles, so \( m\angle CDE = m\angle CAB \).

Step2 (corrected): Set up the equation

We have \( - 8x + 93=99 - 9x \)

Step3 (corrected): Solve for \( x \)

Add \( 9x \) to both sides:

$$ -8x + 9x+93=99 - 9x+9x $$

$$ x + 93=99 $$

Subtract 93 from both sides:

$$ x=99 - 93 $$

$$ x = 6 $$

Step4: Find \( m\angle CDE \)

Substitute \( x = 6 \) into the expression for \( m\angle CDE \): \( m\angle CDE=99-9x \)

$$ m\angle CDE=99-9\times6 $$

$$ m\angle CDE=99 - 54 $$

$$ m\angle CDE = 45 $$

Answer:

\( 45 \)