Question

First find two lines (other than $\overleftrightarrow{YD}$and $\overleftrightarrow{YF}$) that are perpendicular. Then write a two-column proof that the lines are perpendicular.

Given: $\overleftrightarrow{YD}\perp \overleftrightarrow{YF}$;

$m\angle 7=m\angle 5$;

$m\angle 8=m\angle 6$

Prove: ? ?

Short answer

The required lines are$\overline{XD}\perp \overline{XF}$ and the following table represents the required answers:

Statements	Reasons
1.$\overleftrightarrow{YD}\perp \overleftrightarrow{YF}$	1. Given
2.$\angle 5+\angle 6=90°$	2. Defintion of$\perp$lines
3. $\angle 5=\angle 7$and$\angle 6=\angle 8$	3. Given
4.$\angle 7+\angle 8=90°$	4. Substitution property
5.$\angle DXF=90°$	5. Observation from given figure
6.$\overleftrightarrow{XD}\perp \overleftrightarrow{XF}$	6. Defintion of$\perp$lines

Step-by-step solution

Step 1. Use the definition of perpendicular lines.

The definition of perpendicular lines states that the relation between two lines is perpendicular when the measurement of angles formed by those lines is equal to 90°.Using this definition determine the angle having the measurement of 90° that is ∠DYF.

Step 2. Observe the ∠DYF.

∠DYF consists of two angles that are ∠5 and ∠6 also the measurement of these angles is equal to 90° if they are added. So by using this information make an equation.

$\angle 5+\angle 6=90°$

Step 3. Determine the addition of ∠7 and ∠8.

Refer to the measurement of provided angles so ∠5 can be replaced with ∠7 and ∠6 can be replaced with ∠8 into the obtained equation so rewrite the equation according to it.

$\angle 7+\angle 8=90°$

Step 4. Observe the obtained equation.

As angles ∠3 and ∠1 are included in ∠DXF which means that ∠DXF is equal to 90° so according to the definition of perpendicular lines the lines XD and XF are having perpendicular relation between them because ∠DXF is formed by the lines XD and XF.

Step 5. Make the required table by using each step of proving.

Make two columns of statement and reasons after it fill each row according to the step of proving.

Statements	Reasons
1.$\overleftrightarrow{YD}\perp \overleftrightarrow{YF}$	1. Given
2.$\angle 5+\angle 6=90°$	2. Defintion of$\perp$lines
3.$\angle 5=\angle 7$ and$\angle 6=\angle 8$	3. Given
4.$\angle 7+\angle 8=90°$	4. Substitution property
5.$\angle DXF=90°$	5. Observation from given figure
6.$\overleftrightarrow{XD}\perp \overleftrightarrow{XF}$	6. Defintion of $\perp$lines