Question

What can you conclude from the information given?

Given: $\overrightarrow{AD}\perp \overleftrightarrow{AC}$; $\overrightarrow{CE}\perp \overleftrightarrow{AC}$; $m\angle 1=m\angle 4$

Short answer

The conclusion is $m\angle 2=m\angle 3$.

Step-by-step solution

Step 1. Definition of perpendicular lines.

The definition of perpendicular lines states that the relation between two lines is perpendicular when the property of angles formed by those lines is represented by congruent adjacent angles.

Step 2. Observe the diagram and use the given relation between the lines to determine the angles having the measurement of 90°.

The ∠DAC and ∠ECA represent the right angle so their value will be equal to 90°.

Step 3. Determine the addition that is related to ∠1 and ∠2

The ∠DAC comprises the angles ∠1 and ∠2 as is a right angle so the addition of ∠1 and ∠2 will be equal to 90°.

$\angle 1+\angle 2=90°$

Step 4. Determine the addition that is related to ∠3 and ∠4

The ∠ECA comprises the angles ∠3 and ∠4 as ∠ECA is a right angle so the addition of ∠3 and ∠4 will be equal to 90°.

$\angle 3+\angle 4=90°$

Step 5. Make the conclusion.

According to the given relation between the angles and the property of∠1,∠2,∠3 and∠4 that is represented by congruent adjacent angles, the value of$m\angle 2$ and$m\angle 3$ will be the same. Make an equation that represents the conclusion.

$m\angle 2=m\angle 3$