Question

Copy everything shown and complete the proof of theorem 2-7 for the case where two angles are supplements of the same angle.

Given:∠1 and∠5 are supplementary;∠3 and∠5 are supplementary.

Prove: $\angle 1\cong \angle 3$

Short answer

The complete proof table is:

Statements	Reasons
∠1 and∠5 are supplementary. ∠3 and∠5 are supplementary.	GIven
$m\angle 1+m\angle 5=180$;$m\angle 3+m\angle 5=180$	As,∠1 and∠5 are supplementary angles and ∠3 and ∠5 are supplementary angles
$m\angle 1+m\angle 5=m\angle 3+m\angle 5$	Substitution property
$m\angle 5=m\angle 5$	Reflexive property
$m\angle 1=m\angle 3$or$\angle 1\cong \angle 3$	Solve the equation $m\angle 1+m\angle 5=m\angle 3+m\angle 5$.

Step-by-step solution

Step 1. Observe the given diagram.

The given diagram is:

Step 2. Description of step.

It is being given that∠1 and∠5 are supplementary and∠3 and∠5 are supplementary.

As, and are supplementary angles, therefore the sum of the angles∠1 and∠5 is 180°.

Therefore, $m\angle 1+m\angle 5=180$.

As,∠3 and∠5 are supplementary angles, therefore the sum of the angles∠3 and∠5 is 180°.

Therefore, $m\angle 3+m\angle 5=180$.

Now, by using the substitution property it can be obtained that:

$m\angle 1+m\angle 5=m\angle 3+m\angle 5$

Therefore, it can be obtained that:

$$\begin{gathered} m\angle 1+m\angle 5=m\angle 3+m\angle 5 \\ m\angle 1=m\angle 3 \\ \Rightarrow \angle 1\cong \angle 3 \end{gathered}$$

Hence proved.

Step 3. Complete the proof table.

Statements	Reasons
∠1 and∠5 are supplementary. ∠3 and∠5 are supplementary.	GIven
$m\angle 1+m\angle 5=180$;$m\angle 3+m\angle 5=180$	As,∠1 and∠5 are supplementary angles and ∠3 and∠5 are supplementary angles
$m\angle 1+m\angle 5=m\angle 3+m\angle 5$	Substitution property
$m\angle 5=m\angle 5$	Reflexive property
$m\angle 1=m\angle 3$or$\angle 1\cong \angle 3$	Solve the equation $m\angle 1+m\angle 5=m\angle 3+m\angle 5$.