Question

Copy and complete the proof of Theorem 2-5: If two lines form congruent adjacent angles then the lines are perpendicular.

Given:$\angle 1\cong \angle 2$

Prove: $l\perp n$

Proof:

Statements	Reasons
1. $\angle 1\cong \angle 2$, or$m\angle 1=m\angle 2$	1. _______
2. $m\angle 1+m\angle 2=180$.	2._______
3. $m\angle 1+m\angle 2=180$or $2m\angle 2=180$.	3._______
4. $m\angle 2=90$	4._______
5._______	5. Definition of$\perp$lines

Short answer

The following table represents the required answers:

Statements	Reasons
1. $\angle 1\cong \angle 2$, or$m\angle 1=m\angle 2$	1.Given
2.$m\angle 1+m\angle 2=180$	2. Linear pair of angles
3. $m\angle 1+m\angle 2=180$or$2m\angle 2=180$	3. Substitution property
4.$m\angle 2=90$	4. Division property of equality
5.$l\perp n$	5. Definition of $\perp$lines

Step-by-step solution

Step 1. Fill in the first blank of reasons.

The first statement falls in the category of given so the reason for this will be ‘given’.

Step 2. Fill in the second blank of reasons.

According to the provided information, the$\angle 1$ and$\angle 2$ are related with a straight line and a straight line contains adjacent angles which means that the sum of their measurement will be equal to 180° so fill the second blank with ‘Linear pair of angles’.

Step 3. Fill the third blank of reasons.

As each angle is having an equal measurement which means the$\angle 1$can be replaced with$\angle 2$. Fill the third blank reason with ‘Substitution property’ because the angles are satisfying the substitution property.

Step 4. Fill in the first blank of statements.

According to the fourth statement, both the angles are right angles which means that the relation between the lines is perpendicular. So fill the first blank of the statement with$l\perp n$.

Step 5. Make the complete table.

Using all the obtained values for their respective blanks complete the table.

Statements	Reasons
1. $\angle 1\cong \angle 2$, or$m\angle 1=m\angle 2$	1. Given
2.$m\angle 1+m\angle 2=180$	2. Linear pair of angles
3. $m\angle 1+m\angle 2=180$or$2m\angle 2=180$	3. Substitution property
4.$m\angle 2=90$	4. Division property of equality
5.$l\perp n$	5. Definition of $\perp$lines