# Chapter 5 - Section 5.3 - Proving Lines Parallel - Exercises - Page 233: 36

We have the following proof is paragraph form: We are told that RS and YZ are parallel, and RU and XZ are parallel. Since parallel lines cut by the same transversal have congruent corresponding angles, this means that angle R and angle are congruent. By the vertical angles theorem, it follows that angle RTS is congruent to angle YTU. Thus, by AA, it follows that triangle RST is similar to triangle YTU. Next, we find that angle Y is congruent to itself by the identity property. In addition, because parallel lines cut by the same transversal have congruent corresponding angles, angles Z and U are congruent. Thus, triangles XYZ and YTU are congruent by AA. Because XYZ is similar to YTU and YTU is similar to RTS, it follows by the transitive property that XYZ and RTS are similar triangles. Thus, $\frac{RS}{ZY} = \frac{RT}{ZX}$, for CSSTP. Cross multiplying gives: $RS \cdot ZX = ZY \cdot RT$.

#### Work Step by Step

Both triangles RST and RST are similar by AA. By hypothesis $\overline{RS} \parallel \overline{YZ}$ And since corresponding angles of two parallel lines cut by a transversal are congruent then $\angle RST = \angle ZXY$, Same thing can be applied for second hypothesis that $\overline{RU} \parallel \overline{XZ}$, then $\angle Z = \angle U$ but also $\angle R = \angle U$ by parallel line theorems, hence using transitive property $\angle R= \angle Z$. We proved that $\triangle RST$ ~ $\triangle ZXY$ by AA. Setting up the proportionality of similar triangles $\frac{RS}{ZY}=\frac{RT}{XZ}$ and by mean extreme property RS . ZX= ZY . RT $\square$

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