Question

complete the proof to make the following flow - chart proof true.
given: (mangle pks=manglefill in 1)
(mangle pks + manglefill in 2=mangle pkw) (angle addition)
(manglefill in 3+mangle wkr=mangle skr) (angle addition postulate)
(manglefill in 4+mangle skw=mangle skw + mangle wkr) (addition property of equality)
(angle pkw= mangle skr) (substitution property of equality)
(mangle pkwcong mangle skr) (definition of congruence)
fill in 1:
fill in 2:
fill in 3:
fill in 4:

Explanation:

Step1: Analyze given equal - angle statement

The first statement $m\angle PKS=m\angle(\text{FILL IN 1})$ is given. Since there is no other information to start with, and usually in angle - related proofs, we might have vertical angles or congruent angles given. But looking at the overall proof structure, if we assume the most basic case of angle equality based on the context, and considering the next step is angle addition, a likely candidate is an angle that is equal to $\angle PKS$. Let's assume it's an angle that is part of the larger angle relationships. Since we are dealing with angle addition later, and we know that $\angle PKS$ and another angle combine to form $\angle PKW$, a reasonable fill - in is $\angle SKW$. Because if $m\angle PKS = m\angle SKW$, it can be used in the subsequent angle - addition steps.

Step2: Apply angle - addition postulate

The angle - addition postulate states that if we have two adjacent angles, the measure of the larger angle is the sum of the measures of the two smaller adjacent angles. For $\angle PKW$, we know that $m\angle PKS+m\angle(\text{FILL IN 2})=m\angle PKW$. Since $\angle PKS$ and $\angle SKW$ are adjacent and form $\angle PKW$, $\text{FILL IN 2}$ is $\angle SKW$.

Step3: Apply angle - addition postulate again

For $\angle SKR$, we know that $m\angle(\text{FILL IN 3})+m\angle WKR=m\angle SKR$. Since $\angle SKW$ and $\angle WKR$ are adjacent and form $\angle SKR$, $\text{FILL IN 3}$ is $\angle SKW$.

Step4: Use addition property of equality

We have $m\angle(\text{FILL IN 4})+m\angle SKW=m\angle SKW + m\angle WKR$. From the previous steps, we know that we are building up the angle - equality relationships. Since we already established that $m\angle PKS=m\angle SKW$, $\text{FILL IN 4}$ is $\angle PKS$.

Answer:

FILL IN 1: $\angle SKW$
FILL IN 2: $\angle SKW$
FILL IN 3: $\angle SKW$
FILL IN 4: $\angle PKS$