Question

the diagram shows $\triangle qtr$ and $\triangle uts$. in the diagram, $\frac{qt}{tr} = \frac{ut}{ts}$. complete the proof that $us \parallel qr$.
statement | reason
1 $\frac{qt}{tr} = \frac{ut}{ts}$ | given
2 $\angle t \cong \angle t$ | reflexive property of congruence
3 $\triangle qtr \sim \triangle uts$ |
4 $\angle ust = $
5 $us \parallel qr$ | if two lines are cut by a transversal and the corresponding angles are congruent, then the lines are parallel.

Explanation:

For Statement 3: $\triangle QTR \sim \triangle UTS$

Step 1: Identify Given Proportions and Angle

We know from Statement 1 that $\frac{QT}{TR}=\frac{UT}{TS}$ (rewriting $\frac{QT}{TR}=\frac{UT}{RS}$ as $\frac{QT}{TR}=\frac{UT}{TS}$ assuming typo, or using the given $\frac{QT}{TR}=\frac{UT}{RS}$ with $RS = TS$? Wait, no, the angle $\angle T$ is common (Statement 2: $\angle T\cong\angle T$ by Reflexive Property). So we have two sides in proportion and included angle equal.

Step 2: Apply SAS Similarity Criterion

The Side - Angle - Side (SAS) similarity criterion states that if two sides of one triangle are proportional to two sides of another triangle and the included angles are equal, then the triangles are similar. Here, in $\triangle QTR$ and $\triangle UTS$, $\frac{QT}{UT}=\frac{TR}{TS}$ (from $\frac{QT}{TR}=\frac{UT}{TS}$ cross - multiplied: $QT\times TS=UT\times TR$ which implies $\frac{QT}{UT}=\frac{TR}{TS}$) and $\angle T\cong\angle T$ (included angle). So by SAS similarity, $\triangle QTR\sim\triangle UTS$.

Step 1: Use Similar Triangles Corresponding Angles

Since $\triangle QTR\sim\triangle UTS$ (from Statement 3), the corresponding angles of similar triangles are congruent. In similar triangles $\triangle QTR$ and $\triangle UTS$, $\angle UST$ corresponds to $\angle QRT$.

Step 2: Identify Corresponding Angles

In $\triangle QTR$ and $\triangle UTS$, the order of the letters in the similarity statement $\triangle QTR\sim\triangle UTS$ means that $\angle Q$ corresponds to $\angle U$, $\angle T$ corresponds to $\angle T$, and $\angle R$ (i.e., $\angle QRT$) corresponds to $\angle S$ (i.e., $\angle UST$). So $\angle UST\cong\angle QRT$.

Step 1: Use Corresponding Angles Postulate

If two lines are cut by a transversal and the corresponding angles are congruent, then the lines are parallel. We know from Statement 4 that $\angle UST\cong\angle QRT$. Here, the transversal is $TR$ (or $QR$ and $US$ are cut by $TR$). $\angle UST$ and $\angle QRT$ are corresponding angles.

Step 2: Apply Parallel Lines Criterion

Since $\angle UST\cong\angle QRT$ (corresponding angles), by the Corresponding Angles Postulate, $US\parallel QR$.

Answer:

The reason for $\triangle QTR\sim\triangle UTS$ is the SAS (Side - Angle - Side) Similarity Criterion (since two sides are in proportion $\frac{QT}{TR}=\frac{UT}{TS}$ and the included angle $\angle T$ is congruent to itself).

For Statement 4: $\angle UST\cong$?