Question

Copy everything shown and write a two-column proof.

Given: $m\angle 1=m\angle 2$; $m\angle 3=m\angle 4$.

Prove:$\overrightarrow{YS}\perp \overleftrightarrow{XZ}$

Short answer

The two-column proof is:

Statements	Reasons
$m\angle 1=m\angle 2$	Given
$m\angle 3=m\angle 4$	Given
$m\angle 1+m\angle 2+m\angle 3+m\angle 4=180°$	The angles ∠3,∠1, ∠2 and ∠4 form the straight angle
$m\angle 1+m\angle 1+m\angle 3+m\angle 3=180°$	Substitution property
$2\left(m\angle 1+m\angle 3\right)=180°$	Combine the like terms
$m\angle 1+m\angle 3=90°$	Divide both sides by 2.
$m\angle SYX=90°$	Complementary angles
$\overrightarrow{YS}\perp \overleftrightarrow{XZ}$	Definition of perpendicular lines

Step-by-step solution

Step 1. Observe the diagram.

The given diagram is:

Step 2. Description of step.

It is being given that $m\angle 1=m\angle 2$and $m\angle 3=m\angle 4$.

From the given diagram it can be noticed that angles ∠3, ∠1, ∠2 and ∠4 form the straight angle.

Therefore, $m\angle 1+m\angle 2+m\angle 3+m\angle 4=180°$.

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

$$\begin{gathered} m\angle 1+m\angle 1+m\angle 3+m\angle 3=180° \\ 2\left(m\angle 1+m\angle 3\right)=180° \\ m\angle 1+m\angle 3=\frac{180°}{2} \\ m\angle 1+m\angle 3=90° \end{gathered}$$

Therefore, $m\angle SYX=90°$.

Therefore, by using the definition of perpendicular lines it can be said that $\overrightarrow{YS}\perp \overleftrightarrow{XZ}$.

Hence proved.

Step 3. Description of step.

Statements	Reasons
$m\angle 1=m\angle 2$	Given
$m\angle 3=m\angle 4$	Given
$m\angle 1+m\angle 2+m\angle 3+m\angle 4=180°$	The angles∠3,∠1,∠2 and∠4forms the straight angle
$m\angle 1+m\angle 1+m\angle 3+m\angle 3=180°$	Substitution property
$2\left(m\angle 1+m\angle 3\right)=180°$	Combine the like terms
$m\angle 1+m\angle 3=90°$	Divide both sides by 2.
$m\angle SYX=90°$	Complementary angles
$\overrightarrow{YS}\perp \overleftrightarrow{XZ}$	Definition of perpendicular lines