Question

The two triangles shown are congruent. Complete.

a. $\mathrm{\Delta }STO\cong$? .

b. $\angle S\cong$? because ?.

c. $\angle SO\cong$? because ? .

Then point O is the midpoint of? .

d. $\angle T\cong$? because ?.

Then $\overline{ST}\parallel \overline{RK}$because ?.

Short answer

a. $\mathrm{\Delta }STO\cong \underset{¯}{\mathrm{\Delta }KRO}$

b. $\angle S\cong \underset{¯}{\angle K}$ because when two triangles are congruent, then its corresponding parts are also congruent.

c. $\overline{SO}\cong \underset{¯}{\overline{OK}}$ because when two triangles are congruent, then its corresponding parts are also congruent.

Then $O$is the midpoint of $\underset{¯}{\overline{SK}}$.

d. $\angle T\cong \underset{¯}{\angle R}$ because when two triangles are congruent, then its corresponding parts are also congruent.

Then$\overline{ST}\parallel \overline{RK}$ because alternate interior angles cut by a transversal are equal, that is,$\underset{¯}{\angle S\cong \angle K}$ and $\underset{¯}{\angle T\cong \angle R}$.

Step-by-step solution

Part a. Step 1. Consider the diagram.

Here, the two triangles are congruent.

Part a. Step 2. State the explanation.

As, the triangles are congruent, $\mathrm{\Delta }STO$ is congruent to $\mathrm{\Delta }KRO$.

Part a. Step 3. State the conclusion.

Therefore, $\mathrm{\Delta }STO\cong \mathrm{\Delta }KRO$.

Part b. Step 1. Consider the diagram.

Here, the two triangles are congruent.

Part b. Step 2. State the explanation.

As $\mathrm{\Delta }STO\cong \mathrm{\Delta }KRO$, its corresponding parts are also congruent.

Part b. Step 3. State the conclusion.

Therefore, $\angle S\cong \angle K$.

Part c. Step 1. Consider the diagram.

Here, the two triangles are congruent.

Part c. Step 2. State the explanation.

As $\Delta STO\cong \Delta KRO$, its corresponding parts are also congruent.

Part c. Step 3. State the conclusion.

Therefore, $\overline{SO}\cong \overline{OK}$and as $\overline{SO}=\overline{OK}$, O is the midpoint of $\overline{SK}$ since midpoint divides a line segment into two equal parts.

Part d. Step 1. Consider the diagram.

Here, the two triangles are congruent.

Part d. Step 2. State the explanation.

As $\mathrm{\Delta }STO\cong \mathrm{\Delta }KRO$, its corresponding parts are also congruent.

Part d. Step 3. State the conclusion.

Therefore, $\angle T\cong \angle R$.

$\overline{ST}$is parallel to $\overline{RK}$because alternate interior angles cut by a transversal are equal, that is, $\angle S\cong \angle K$and $\angle T\cong \angle R$.