Question

(1) complete the proof that $m\angle h + m\angle i = 90^\circ$.

statement	reason
2. $m\angle j = 90^\circ$
3. $m\angle h + m\angle i + m\angle j = 180^\circ$
4. $m\angle h + m\angle i + 90^\circ = 180^\circ$
5. $m\angle h + m\angle i = 90^\circ$

Explanation:

Step1: Identify the first reason

From the diagram, $\angle J$ is a right angle, so $\overline{HJ} \perp \overline{IJ}$ by the definition of a right angle (or perpendicular lines form right angles). So the reason for statement 1 is "Definition of perpendicular lines (or right angle)".

Step2: Reason for statement 2

If two lines are perpendicular, the angle between them is $90^\circ$. So since $\overline{HJ} \perp \overline{IJ}$, $m\angle J = 90^\circ$ by the definition of perpendicular lines (or the measure of a right angle is $90^\circ$).

Step3: Reason for statement 3

The sum of the interior angles of a triangle is $180^\circ$. So for $\triangle HIJ$, $m\angle H + m\angle I + m\angle J = 180^\circ$ by the Triangle Angle - Sum Theorem.

Step4: Reason for statement 4

We substitute $m\angle J = 90^\circ$ (from statement 2) into the equation from statement 3. So this is the Substitution Property of Equality.

Step5: Reason for statement 5

We solve the equation $m\angle H + m\angle I + 90^\circ = 180^\circ$ for $m\angle H + m\angle I$. Subtract $90^\circ$ from both sides: $m\angle H + m\angle I=180^\circ - 90^\circ = 90^\circ$. This is the Subtraction Property of Equality (subtracting $90^\circ$ from both sides of the equation).

Answer:

1. Reason: Definition of perpendicular lines (or Right angle definition)
2. Reason: If two lines are perpendicular, the angle between them is \(90^\circ\) (or Definition of right angle)
3. Reason: Triangle Angle - Sum Theorem
4. Reason: Substitution Property of Equality
5. Reason: Subtraction Property of Equality (subtracting \(90^\circ\) from both sides)

(To fill in the table, the reasons are as above for each statement respectively.)