Question

given the diagram below and the following statements:

• $\angle hod \cong \angle zur$
• $\angle doc \cong \angle ruj$

prove that $m\angle hod + m\angle doc = m\angle zuj$.

Explanation:

Step1: Recall Angle Addition Postulate

The Angle Addition Postulate states that if a point lies in the interior of an angle, the sum of the measures of the two smaller angles formed is equal to the measure of the larger angle. For $\angle ZUJ$, since $R$ is a point in its interior, we have $m\angle ZUR + m\angle RUJ = m\angle ZUJ$.

Step2: Use Congruent Angles Definition

Given $\angle HOD \cong \angle ZUR$ and $\angle DOC \cong \angle RUJ$, by the definition of congruent angles, their measures are equal. So $m\angle HOD = m\angle ZUR$ and $m\angle DOC = m\angle RUJ$.

Step3: Substitute the Congruent Angle Measures

Substitute $m\angle HOD$ for $m\angle ZUR$ and $m\angle DOC$ for $m\angle RUJ$ in the equation $m\angle ZUR + m\angle RUJ = m\angle ZUJ$. We get $m\angle HOD + m\angle DOC = m\angle ZUJ$.

Answer:

To complete the proof:

1. The first blank (for $m\angle ZUR + m\angle RUJ = m\angle ZUJ$) is filled by the Angle Addition Postulate.
2. The second blank (for $m\angle DOC + m\angle ZUR = m\angle ZUJ$ transformed to $m\angle HOD + m\angle DOC = m\angle ZUJ$) uses substitution of $m\angle HOD$ for $m\angle ZUR$ (from $\angle HOD \cong \angle ZUR$) and $m\angle DOC$ for $m\angle RUJ$ (from $\angle DOC \cong \angle RUJ$), leading to $m\angle HOD + m\angle DOC = m\angle ZUJ$.

Final logical flow:

• $m\angle ZUR + m\angle RUJ = m\angle ZUJ$ (Angle Addition Postulate)
• $m\angle HOD = m\angle ZUR$ and $m\angle DOC = m\angle RUJ$ (Definition of congruent angles)
• Substitute to get $m\angle HOD + m\angle DOC = m\angle ZUJ$ (Substitution Property of Equality).