Questions: Let AB and CD be parallel, and let AD and BC intersect at E. Prove that a * y = x * b. In order for a * y = x * b, what must be true about the two triangles? A. The two triangles must have three congruent angles. B. The two triangles must have equal sides and congruent angles. C. The two triangles must have at least two sides of equal length. D. The two triangles must be similar.

Let AB and CD be parallel, and let AD and BC intersect at E. Prove that a * y = x * b.

In order for a * y = x * b, what must be true about the two triangles?
A. The two triangles must have three congruent angles.
B. The two triangles must have equal sides and congruent angles.
C. The two triangles must have at least two sides of equal length.
D. The two triangles must be similar.

Transcript text: Let $\overline{\mathrm{AB}}$ and $\overline{\mathrm{CD}}$ be parallel, and let $\overline{\mathrm{AD}}$ and $\overline{\mathrm{BC}}$ intersect at E . Prove that $a \cdot y=x \cdot b$. In order for $a \cdot y=x \cdot b$, what must be true about the two triangles? A. The two triangles must have three congruent angles. B. The two triangles must have equal sides and congruent angles. C. The two triangles must have at least two sides of equal length. D. The two triangles must be similar.

Solution

Solution Steps

Step 1: Identify the Given Information

We are given that $AB$ and $CD$ are parallel, and $AD$ and $BC$ intersect at $E$. We need to prove that $a \cdot y = x \cdot b$.

Step 2: Use the Properties of Parallel Lines

Since $AB \parallel CD$, the corresponding angles formed by the transversal lines $AD$ and $BC$ are equal. This means:

$\angle AEB = \angle CED$
$\angle BAE = \angle DCE$

Step 3: Establish Similarity of Triangles

From the angle properties, we can deduce that $\triangle AEB$ and $\triangle CED$ are similar by the AA (Angle-Angle) similarity criterion.

Step 4: Apply the Proportionality of Similar Triangles

Since $\triangle AEB \sim \triangle CED$, the corresponding sides of similar triangles are proportional. Therefore: \[ \frac{AE}{EC} = \frac{AB}{CD} \quad \text{and} \quad \frac{BE}{ED} = \frac{AB}{CD} \]

Step 5: Relate the Sides to the Given Variables

From the similarity and proportionality, we can write: \[ \frac{a}{x} = \frac{y}{b} \]

Step 6: Cross-Multiply to Prove the Required Equation

Cross-multiplying the above proportion gives: \[ a \cdot y = x \cdot b \]