Question

Given: $\angle 2\cong \angle 3$.

a. What can you deduce?

b. Explain how you would prove your conclusion.

Short answer

a.It can be deduced that the lines AB and CD are parallel.

b. It is being given that$\angle 2\cong \angle 3$.

From the given diagram it can be noticed that the angles $\angle \mathbf{AGE}$and $\angle \mathbf{BGH}$are the vertically opposite angles.

Therefore, $\angle \mathbf{AGE}\cong \angle \mathbf{BGH}$.

Therefore, it can be obtained that:

$\begin{array}{c}\angle \mathbf{AGE}\cong \angle \mathbf{BGH}\\ \angle \mathbf{1}\cong \angle \mathbf{2}\end{array}$

As, $\angle \mathbf{1}\cong \angle \mathbf{2}$and $\angle \mathbf{2}\cong \angle \mathbf{3}$, therefore by using the transitive property of congruence it can be said that$\angle \mathbf{1}\cong \angle \mathbf{3}$

As, $\angle \mathbf{1}\cong \angle \mathbf{3}$, therefore $\angle \mathbf{AGE}\cong \angle \mathbf{CHG}$.

From the figure, it can be noticed that the angles$\angle \mathbf{AGE}$ and$\angle \mathbf{CHG}$ are corresponding angles.

If the two lines are parallel and there is a transversal then the corresponding angles are congruent.

Therefore, as corresponding angles$\angle \mathbf{AGE}$ and$\angle \mathbf{CHG}$ are congruent.

Therefore, the lines AB and CD are parallel.

Hence proved.

Step-by-step solution

Part a. Step 1. Label the given diagram.

The labelled diagram is:

Part a. Step 2. Description of step.

It is being given that $\angle 2\cong \angle 3$.

Therefore, $\angle BGH\cong \angle CHG$.

From the given diagram it can be noticed that the angles $\angle BGH$and$\angle CHG$ are alternate interior angles.

If the two lines are parallel and there is a transversal then the alternate interior angles are congruent.

Therefore, as alternate interior angles$\angle BGH$ and$\angle CHG$ are congruent.

Therefore, the lines AB and CD are parallel.

Part a. Step 3. Description of step.

Therefore, it can be deduced that the lines AB and CD are parallel.

Part b. Step 1. Label the given diagram.

The labelled diagram is:

Part b. Step 2. Description of step.

It is being given that$\angle 2\cong \angle 3$.

From the given diagram it can be noticed that the angles$\angle AGE$ and$\angle BGH$ are the vertically opposite angles.

Therefore, $\angle AGE\cong \angle BGH$.

Therefore, it can be obtained that:

$\begin{array}{c}\angle AGE\cong \angle BGH\\ \angle 1\cong \angle 2\end{array}$

Part b. Step 3. Description of step.

As,$\angle 1\cong \angle 2$ and $\angle 2\cong \angle 3$, therefore by using the transitive property of congruence it can be said that$\angle 1\cong \angle 3$

As, $\angle 1\cong \angle 3$, therefore $\angle AGE\cong \angle CHG$.

From the figure it can be noticed that the angles$\angle AGE$ and$\angle CHG$ are corresponding angles.

If the two lines are parallel and there is a transversal then the corresponding angles are congruent.

Therefore, as corresponding angles $\angle AGE$and $\angle CHG$are congruent.

Therefore, the lines AB and CD are parallel.

Hence proved.