Question

)) ( overline{fg} perp overline{eg} ). complete the proof that ( mangle e + mangle f = 90^circ ).

statement	reason
2 ( mangle g = 90^circ )	aa similarity, aas, additive property of angle measure, additive property of length, all right angles are congruent, alternate exterior angles theorem
3 ( mangle e + mangle f + mangle g = 180^circ )
4 ( mangle e + mangle f + 90^circ = 180^circ )
5 ( mangle e + mangle f = 90^circ )

Explanation:

Step1: Recall Perpendicular Angles

When two lines are perpendicular, the angle between them is \( 90^\circ \). So for \( \overline{FG} \perp \overline{EG} \), the reason is the definition of perpendicular lines (which implies the angle formed is a right angle, \( 90^\circ \)). But looking at the dropdown for statement 1, we need to pick the correct reason. Wait, actually, let's check each step:

Wait, the first statement is \( \overline{FG} \perp \overline{EG} \), the reason for that (if we were filling it) would be "Definition of Perpendicular Lines" (since perpendicular lines form a right angle). But in the given dropdown for statement 2, the reason for \( m\angle G = 90^\circ \) is because \( \overline{FG} \perp \overline{EG} \), so the reason for statement 1 (if we consider the first blank) – wait, maybe the question is about completing the proof, like for statement 1, the reason? Wait, no, the problem says "Complete the proof that \( m\angle E + m\angle F = 90^\circ \)". Let's go step by step.

Step2: Analyze Statement 1 Reason

Statement 1: \( \overline{FG} \perp \overline{EG} \) – the reason here, but looking at the dropdown for statement 2, maybe the first blank (statement 1's reason) is not shown, but actually, let's check the steps. Wait, the table has statements 1 - 5, with reasons for some. Wait, maybe the question is about the reason for statement 1? Wait, no, the user's problem is to complete the proof. Let's re-express:

1. \( \overline{FG} \perp \overline{EG} \) – Reason: Definition of Perpendicular Lines (since perpendicular lines intersect at \( 90^\circ \), so \( \angle G \) is \( 90^\circ \)).

2. \( m\angle G = 90^\circ \) – Reason: If two lines are perpendicular, then they form a right angle (so from statement 1, \( \overline{FG} \perp \overline{EG} \) implies \( \angle G = 90^\circ \), so the reason here would be "Definition of Perpendicular Lines" or "Perpendicular lines form a right angle". But in the dropdown, the options include "All right angles are congruent" – no, wait, for statement 2, \( m\angle G = 90^\circ \) because \( \overline{FG} \perp \overline{EG} \), so the reason is the definition of perpendicular lines (which is that they form a right angle, \( 90^\circ \)).

Wait, maybe the first blank (statement 1's reason) is missing, but the problem is to complete the proof. Let's check the steps:

- Step 1: \( \overline{FG} \perp \overline{EG} \) – Reason: (should be "Definition of Perpendicular Lines" or similar, but in the dropdown, maybe the option is not there, but wait the user's image shows a dropdown for statement 1's reason? Wait, the image has a dropdown for statement 1 (the first reason box) and then for statement 5. Wait, maybe the question is about the reason for statement 1. Let's see the options in the dropdown for statement 1 (the first box): the options include "All right angles are congruent" – no, that's for congruent right angles. Wait, no, the correct reason for \( \overline{FG} \perp \overline{EG} \) leading to \( m\angle G = 90^\circ \) is that perpendicular lines form a right angle, so the reason for statement 2 (\( m\angle G = 90^\circ \)) is "If two lines are perpendicular, then the angle between them is \( 90^\circ \)" (which is the definition of perpendicular lines). But in the dropdown, the options for statement 2's reason (wait, no, statement 2's reason is currently "AA Similarity" which is wrong. Wait, the user's table has:

Statement 2: \( m\angle G = 90^\circ \), Reason: AA Similarity (wrong). So we need to correct that.

Let's redo the proof:

1. \( \overline{FG} \perp \overline{EG} \) – Reason: Given (or Definition of Perpendicular Lines)
2. \( m\angle G = 90^\circ \) – Reason: Definition of Perpendicular Lines (since perpendicular lines form a right angle)
3. \( m\angle E + m\angle F + m\angle G = 180^\circ \) – Reason: Triangle Angle Sum Theorem (sum of angles in a triangle is \( 180^\circ \))
4. \( m\angle E + m\angle F + 90^\circ = 180^\circ \) – Reason: Substitution (replace \( m\angle G \) with \( 90^\circ \) from statement 2)
5. \( m\angle E + m\angle F = 90^\circ \) – Reason: Subtraction Property of Equality (subtract \( 90^\circ \) from both sides)

But in the dropdown options for statement 2's reason, the correct option should be related to perpendicular lines forming a right angle. Looking at the dropdown options: "All right angles are congruent" – no, that's for congruence. Wait, the options include "All right angles are congruent" – no, the correct reason for \( m\angle G = 90^\circ \) is that \( \overline{FG} \perp \overline{EG} \), so the reason is "Definition of Perpendicular Lines" (which is not in the dropdown? Wait, the dropdown options given are: AA Similarity, AAS, Additive Property of Angle Measure, Additive Property of Length, All right angles are congruent, Alternate Exterior Angles Theorem. Wait, maybe the first blank (statement 1's reason) is "Given" (but it's not in the dropdown). Wait, maybe the problem is about the reason for statement 5? No, statement 5 is \( m\angle E + m\angle F = 90^\circ \), reason should be Subtraction Property of Equality, but in the dropdown, maybe "Additive Property of Angle Measure" – no. Wait, let's check the steps again.

Wait, the sum of angles in a triangle is \( 180^\circ \) (statement 3, reason: Triangle Angle Sum Theorem, but in the dropdown, "Additive Property of Angle Measure" – maybe that's the reason for statement 3? Wait, statement 3: \( m\angle E + m\angle F + m\angle G = 180^\circ \), reason is "Additive Property of Angle Measure" (since the sum of the measures of the angles in a triangle is the sum of their individual measures, which is the additive property). Then statement 4: substitute \( m\angle G = 90^\circ \) (from statement 2), so reason is "Substitution" (but not in dropdown). Wait, the dropdown for statement 5's reason: the options include "Alternate Exterior Angles Theorem" – no. Wait, maybe the correct reason for statement 5 is "Subtraction Property of Equality" (subtract \( 90^\circ \) from both sides of \( m\angle E + m\angle F + 90^\circ = 180^\circ \) to get \( m\angle E + m\angle F = 90^\circ \)). But in the dropdown, maybe the closest is not there, but perhaps the user's question is about the reason for statement 1. Wait, no, the problem says "Complete the proof that \( m\angle E + m\angle F = 90^\circ \)". Let's summarize the correct proof steps:

1. \( \overline{FG} \perp \overline{EG} \) – Reason: Given (or Definition of Perpendicular Lines)
2. \( m\angle G = 90^\circ \) – Reason: If two lines are perpendicular, they form a right angle (so from 1, \( \angle G = 90^\circ \))
3. \( m\angle E + m\angle F + m\angle G = 180^\circ \) – Reason: Triangle Angle Sum Theorem (or Additive Property of Angle Measure, since the sum of angles in a triangle is the sum of their measures)
4. \( m\angle E + m\angle F + 90^\circ = 180^\circ \) – Reason: Substitution (replace \( m\angle G \) with \( 90^\circ \) from step 2)
5. \( m\angle E + m\angle F = 90^\circ \) – Reason: Subtraction Property of Equality (subtract \( 90^\circ \) from both sides)

But in the given dropdown options, for statement 2's reason, the correct option should be related to perpendicular lines forming a right angle, but the options include "All right angles are congruent" – no. Wait, maybe the first blank (statement 1's reason) is "Given" (if it's a given statement), but the dropdown for statement 1 is empty. Alternatively, maybe the question is about the reason for statement 5, which is "Subtraction Property of Equality", but in the dropdown, maybe the option is not there, but perhaps the user made a typo. Wait, the image shows that for statement 5, the reason is partially visible as "Alternate Exterior Angles Theorem" which is wrong. The correct reason is subtracting \( 90^\circ \) from both sides, so using the Subtraction Property of Equality, but if that's not an option, maybe the closest is "Additive Property of Angle Measure" – no. Wait, maybe the problem is simpler: since \( \overline{FG} \perp \overline{EG} \), \( \angle G = 90^\circ \), triangle sum is \( 180^\circ \), so \( \angle E + \angle F = 90^\circ \). So the key steps are using the definition of perpendicular lines (to get \( \angle G = 90^\circ \)), triangle angle sum, and then subtraction.

But maybe the question is about the reason for statement 1: the reason is "Definition of Perpendicular Lines" (since perpendicular lines intersect at \( 90^\circ \), so \( \angle G = 90^\circ \) follows). But in the dropdown, if we have to choose, maybe for statement 2, the reason is "All right angles are congruent" – no, that's for congruence. Wait, no, the correct reason for \( m\angle G = 90^\circ \) is that \( \overline{FG} \perp \overline{EG} \), so the reason is "Definition of Perpendicular Lines" (which is that two lines are perpendicular if and only if they form a right angle, \( 90^\circ \)).

Alternatively, maybe the first blank (statement 1's reason) is "Given" (if it's a given fact), and then statement 2's reason is "Perpendicular lines form a right angle" (but not in dropdown). Wait, the dropdown options for statement 2's reason include "All right angles are congruent" – no, that's not. Maybe the user's problem is to identify the correct reason for a step. Let's assume that the proof is:

1. \( \overline{FG} \perp \overline{EG} \) – Reason: Given (or Definition of Perpendicular Lines)
2. \( m\angle G = 90^\circ \) – Reason: If two lines are perpendicular, the angle between them is \( 90^\circ \) (Definition of Perpendicular Lines)
3. \( m\angle E + m\angle F + m\angle G = 180^\circ \) – Reason: Triangle Angle Sum Theorem (or Additive Property of Angle Measure)
4. \( m\angle E + m\angle F + 90^\circ = 180^\circ \) – Reason: Substitution (from step 2)
5. \( m\angle E + m\angle F = 90^\circ \) – Reason: Subtraction Property of Equality (subtract \( 90^\circ \) from both sides)

But since the dropdown options include "All right angles are congruent" – no, maybe the answer is that the reason for statement 2 is "All right angles are congruent" – no, that's incorrect. Wait, maybe the problem is about the first step: the reason for \( \overline{FG} \perp \overline{EG} \) is "Given" (if it's a given), and then the rest follows. But perhaps the user is asking for the reason to fill in the first blank (statement 1's reason), which is "Definition of Perpendicular Lines" (or "Given"). But since the options are in the dropdown, maybe the correct reason for statement 2 is "All right angles are congruent" – no, that's not. I think the key here is that the sum of angles in a triangle is \( 180^\circ \), so with \( \angle G = 90^\circ \), the other two angles sum to \( 90^\circ \). So the proof uses the definition of perpendicular lines (to get \( \angle G = 90^\circ \)), triangle angle sum, and subtraction.

Answer:

To complete the proof:

1. Reason for \( \overline{FG} \perp \overline{EG} \): Definition of Perpendicular Lines (perpendicular lines form a \( 90^\circ \) angle).
2. Reason for \( m\angle G = 90^\circ \): Follows from \( \overline{FG} \perp \overline{EG} \) (perpendicular lines create a right angle).
3. Reason for \( m\angle E + m\angle F + m\angle G = 180^\circ \): Triangle Angle Sum Theorem (or Additive Property of Angle Measure, as the sum of angle measures in a triangle is \( 180^\circ \)).
4. Reason for \( m\angle E + m\angle F + 90^\circ = 180^\circ \): Substitution (replace \( m\angle G \) with \( 90^\circ \) from Step 2).
5. Reason for \( m\angle E + m\angle F = 90^\circ \): Subtraction Property of Equality (subtract \( 90^\circ \) from both sides of \( m\angle E + m\angle F + 90^\circ = 180^\circ \)).

(If selecting from the dropdown, the correct reason for \( m\angle G = 90^\circ \) is implied by the definition of perpendicular lines, and for the final step, subtraction of \( 90^\circ \) confirms \( m\angle E + m\angle F = 90^\circ \).)

Final conclusion: \( \boldsymbol{m\angle E + m\angle F = 90^\circ} \) is proven using the triangle angle sum theorem and properties of perpendicular lines.