Question

Copy everything shown and write a two-column proof.
Make a diagram showing$\angle PQR$ bisected by role="math" localid="1646848681352" $\overrightarrow{QX}$. Choose a point Y on the ray opposite to $\overrightarrow{QX}$.

Prove:$\angle PQY\cong \angle RQY$

Short answer

The two-column proof is:

Statements	Reasons
Point Y is on the ray opposite to$\overrightarrow{QX}$, $\angle PQR$is bisected by $\overrightarrow{QX}$.	Given
$\angle 1=\angle 2$	$\angle PQR$is bisected by$\overrightarrow{QX}$
$\angle PQY+\angle 1=180$	Definition of supplementary angles
$\angle RQY+\angle 2=180$	Definition of supplementary angles
$\angle PQY+\angle 1=\angle RQY+\angle 2$	Substitution property
$\angle PQY+\angle 1=\angle RQY+\angle 1$	As$\angle 1=\angle 2$
$\angle PQY\cong \angle RQY$	By solving the equation $\angle PQY+\angle 1=\angle RQY+\angle 1$.

Step-by-step solution

Step 1. Draw the diagram.

The diagram depicting the given situation is:

Step 2. Description of step.

It is being given that point Y is on the ray opposite to$\overrightarrow{QX}$and∠PQRis bisected by $\overrightarrow{QX}$.

As,∠PQRis bisected by localid="1646849179080" $\overrightarrow{QX}$, therefore$\angle 1=\angle 2$.

From the diagram, it can be noticed that the angles∠PQY and∠1 forms the linear pair.

Therefore, the sum of the angles∠PQY and∠1 is 180.

That implies, $\angle PQY+\angle 1=180$.

From the diagram, it can be noticed that the angles∠RQY and∠2 forms the linear pair.

Therefore, the sum of the angles∠RQY and∠2 is 180.

That implies, role="math" localid="1646849374087" $\angle PQY+\angle 1=180$.

Therefore, by using the substation property it can be obtained that:

$\angle PQY+\angle 1=\angle RQY+\angle 2$

As, $\angle 1=\angle 2$, therefore it can be obtained that:

$\begin{array}{c}\angle PQY+\angle 1=\angle RQY+\angle 2\\ \angle PQY+\angle 1=\angle RQY+\angle 1\\ \angle PQY=\angle RQY\end{array}$

Therefore, $\angle PQY\cong \angle RQY$.

Hence proved.

Step 3. Write the two column-proof.

Statements	Reasons
Point Y is on the ray opposite to$\overrightarrow{QX}$, $\angle PQR$is bisected by $\overrightarrow{QX}$.	Given
$\angle 1=\angle 2$	$\angle PQR$is bisected by
$\angle PQY+\angle 1=180$	Definition of supplementary angles
$\angle RQY+\angle 2=180$	Definition of supplementary angles
$\angle PQY+\angle 1=\angle RQY+\angle 2$	Substitution property
$\angle PQY+\angle 1=\angle RQY+\angle 1$	As$\angle 1=\angle 2$
$\angle PQY\cong \angle RQY$	By solving the equation $\angle PQY+\angle 1=\angle RQY+\angle 1$.