Question

Can you conclude from the information given for the exercise that $\overrightarrow{XY}\perp \overrightarrow{XZ}$?

$\angle 1$and role="math" localid="1646422063897" $\angle 3$ are congruent and complementary.

Short answer

The relation between the provided lines can be concluded.

Step-by-step solution

Step 1. Determine the sum of all angles.

The position of all the angles is on a straight line which means that the sum of all the angles will be equal to 180°. By using this information make an equation.

$\angle 1+\angle 2+\angle 3+\angle 4=180°$

Step 2. Use the definition of congruent and complementary angles.

Angles having the same measurement are referred as the congruent angles.

Angles having the measurement of 90°when they added are referred as the complementary angles.

Make two equations by using the property of provided angles and the definitions.

$$\begin{gathered} \angle 1=\angle 3 \\ \angle 1+\angle 3=90° \end{gathered}$$

Step 3. Determine the measurement of the provided angles.

Using the properties of provided angles write the value of each angle.

$\begin{array}{c}2\angle 1=2\angle 2=90°\\ \angle 1=\angle 3=45°\end{array}$

Step 4. Determine the value of ∠2

The relation between the lines $\overleftrightarrow{XY}$ and X is perpendicular which means that angles ∠1 and ∠2 represent the complementary angles. So make an equation by using this information after it uses the value of ∠1 and determines the value of ∠2.

$$\begin{gathered} \angle 1+\angle 2=90° \\ 45+\angle 2=90° \\ \angle 2=90°-45° \\ =45° \end{gathered}$$

Step 5. Make the conclusion.

The angles∠2 and∠3 are also representing the complementary angles so make an equation by using this information.

$\angle 2+\angle 3=90°$

The provided figure represents that the position of angles ∠2 and ∠3 is in the middle of $\overrightarrow{XY}$and $\overrightarrow{XZ}$. This represents that these two lines are having a relationship that is perpendicular so the provided relation between the lines can be concluded.