Question

completing measure proofs given the diagram below and the following statements: ∠hod ≅ ∠zur, ∠doc ≅ ∠ruw. prove that m∠hod + m∠doc = m∠zuw. the proof diagram and steps involve angle addition and congruent angle definitions, with boxes to fill in statements related to m∠zur + m∠ruw = m∠zuw, m∠doc + m∠zur = m∠zuw (wait, need to parse the image text better, but the main task is to prove the angle sum using congruent angles and angle addition postulate).

Explanation:

Step1: Identify Given Congruences

We know $\angle HOD \cong \angle ZUR$ and $\angle DOC \cong \angle RUW$. By the definition of congruent angles, $m\angle HOD = m\angle ZUR$ and $m\angle DOC = m\angle RUW$.

Step2: Analyze Angle Addition

From the diagram, $m\angle ZOR = m\angle ZUR + m\angle RUW$ (angle addition postulate). Substituting the congruent angle measures, we get $m\angle ZOR = m\angle HOD + m\angle DOC$. Also, from the other part, $m\angle DOC + m\angle ZUR = m\angle ZU$ (wait, maybe better to track the proof structure). Wait, the goal is to show $m\angle HOD + m\angle DOC = m\angle ZU$. Since $\angle HOD \cong \angle ZUR$ (so $m\angle HOD = m\angle ZUR$) and $\angle DOC \cong \angle RUW$ (so $m\angle DOC = m\angle RUW$), and $m\angle ZU = m\angle ZUR + m\angle RUW$ (angle addition), substituting gives $m\angle ZU = m\angle HOD + m\angle DOC$. So the missing step is using the angle addition postulate and congruent angle substitutions. But looking at the proof tree, the middle box (where $m\angle DOC + m\angle ZUR = m\angle ZU$) comes from angle addition: $m\angle ZU = m\angle ZUR + m\angle RUW$, but since $m\angle DOC = m\angle RUW$ (from $\angle DOC \cong \angle RUW$), we substitute to get $m\angle ZU = m\angle ZUR + m\angle DOC$, which rearranges to $m\angle DOC + m\angle ZUR = m\angle ZU$. Then, since $m\angle HOD = m\angle ZUR$ (from $\angle HOD \cong \angle ZUR$), we substitute $m\angle HOD$ for $m\angle ZUR$ to get $m\angle HOD + m\angle DOC = m\angle ZU$.

Answer:

To complete the proof:

1. Given $\angle HOD \cong \angle ZUR$ and $\angle DOC \cong \angle RUW$, so $m\angle HOD = m\angle ZUR$ and $m\angle DOC = m\angle RUW$ (Definition of Congruent Angles).
2. By the Angle Addition Postulate, $m\angle ZU = m\angle ZUR + m\angle RUW$.
3. Substitute $m\angle DOC$ for $m\angle RUW$ (from step 1) into the angle addition: $m\angle ZU = m\angle ZUR + m\angle DOC$, or $m\angle DOC + m\angle ZUR = m\angle ZU$.
4. Substitute $m\angle HOD$ for $m\angle ZUR$ (from step 1) into the above: $m\angle HOD + m\angle DOC = m\angle ZU$.

The key missing logical step (in the proof tree) is recognizing the angle addition and congruence substitutions to link the angle measures. The final conclusion is $m\angle HOD + m\angle DOC = m\angle ZU$ as required.