Question

a. Suppose $\overrightarrow{\text{SV}}$bisects $\angle RST$and$\overrightarrow{RU}$ bisects $\angle SRT$. State which angles are congruent.

b. State the additional information that will enable a person to deduce that $m\angle VSU=m\angle URV$.

Short answer

a. The congruent angles are:

$\angle TSV,\angle RSV$ and $\angle SRU,\angle TRU$.

b. The additional information required is:

$m\angle VSR=m\angle URS$

Step-by-step solution

Part a. Step 1. Observe the diagram.

$\overrightarrow{SV}$ bisects$\angle RST$ and$\overrightarrow{RU}$ bisects $\angle SRT$ .

Part a. Step 2. State the definition of an angle bisector.

An angle bisector bisects the angle into two congruent angles.

Part a. Step 3. State the conclusion.

As $\overrightarrow{SV}$bisects $\angle RST$,

$\angle TSV\cong \angle RSV$

As $\overrightarrow{RU}$bisects $\angle SRT$,

$\angle SRU\cong \angle TRU$.

Therefore, the congruent angles are $\angle TSV,\angle RSV$and $\angle SRU,\angle TRU$.

Part b. Step 1. Observe the diagram.

$\overrightarrow{SV}$ bisects$\angle RST$ and$\overrightarrow{RU}$ bisects $\angle SRT$ .

Part b. Step 2. State the congruent angles.

The congruent angles are $\angle TSV,\angle RSV$and$\angle SRU,\angle TRU$ as obtained in part (a).

Given:$m\angle VSR=m\angle URV$

Part b. Step 3. State the conclusion.

If two angles are congruent then their measures are also equal.

From part (a), $\angle TSV\cong \angle RSV$, so,

$m\angle TSV=m\angle RSV$ …… (1)

From part (a), $\angle SRU\cong \angle TRU$, so,

$m\angle SRU=m\angle TRU$ …… (2)

Now assume $m\angle VSR=m\angle URS$ …… (3)

From (1), (2) and (3),

$m\angle TSV=m\angle TRU$.

That is,

$m\angle VSU=m\angle URV$

Therefore, the additional information required to deduce $m\angle VSU=m\angle URV$is $m\angle VSR=m\angle URS$.