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what is ed?
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triangle diagram with points a, b, e, c, d. ae = 2x + 4, ed = x + 4, ab = 9, cd = 6. arrows on ab and cd indicate they are parallel.

Explanation:

Step1: Identify Similar Triangles

Triangles \( \triangle ABE \) and \( \triangle DCE \) are similar (by AA similarity, as \( \angle AEB = \angle DEC \) (vertical angles) and \( AB \parallel CD \) implies \( \angle A = \angle D \) and \( \angle B = \angle C \)). So, the ratios of corresponding sides are equal: \( \frac{AB}{CD} = \frac{AE}{ED} \).

Step2: Set Up Proportion

Given \( AB = 9 \), \( CD = 6 \), \( AE = 2x + 4 \), \( ED = x + 4 \). So, \( \frac{9}{6} = \frac{2x + 4}{x + 4} \).

Step3: Simplify Proportion

Simplify \( \frac{9}{6} \) to \( \frac{3}{2} \). So, \( \frac{3}{2} = \frac{2x + 4}{x + 4} \). Cross - multiply: \( 3(x + 4)=2(2x + 4) \).

Step4: Solve for \( x \)

Expand both sides: \( 3x+12 = 4x + 8 \). Subtract \( 3x \) from both sides: \( 12=x + 8 \). Subtract 8 from both sides: \( x = 4 \).

Step5: Find \( ED \)

Substitute \( x = 4 \) into \( ED=x + 4 \). So, \( ED=4 + 4=8 \).

Answer:

8