Question

(f).

1. if ( mangle 3 = 30^circ ), then ( mangle 4 = 150^circ ) \tt \tf
2. if ( mangle 1 = 140^circ ), then ( mangle 4 = 40^circ ) \tt \tf
3. ( angle 2 ) and ( angle 4 ) are congruent \t\t\tt \tf
4. ( mangle 3 + mangle 1 = mangle 4 + mangle 2 ) \t\tt \tf
5. ( angle 3 cong angle 4 ) \t\t\t\t\t\tt \tf
6. ( mangle 3 = 180^circ - mangle 2 ) \t\t\t\tt \tf

Explanation:

To solve these angle - related true - false questions, we assume that the angles are formed by intersecting lines (probably vertical angles and linear pairs, where a linear pair of angles sums up to \(180^{\circ}\) and vertical angles are congruent).

Question 1

Step 1: Analyze the relationship between \(\angle3\) and \(\angle4\)

We assume that \(\angle3\) and \(\angle4\) form a linear pair. By the definition of a linear pair, the sum of the measures of two angles in a linear pair is \(180^{\circ}\), i.e., \(m\angle3 + m\angle4=180^{\circ}\).

Step 2: Calculate \(m\angle4\)

Given \(m\angle3 = 30^{\circ}\), we substitute into the equation \(m\angle4=180^{\circ}-m\angle3\). So \(m\angle4 = 180^{\circ}- 30^{\circ}=150^{\circ}\).
So the statement "If \(m\angle3 = 30^{\circ}\), then \(m\angle4 = 150^{\circ}\)" is True (T).

Question 2

Step 1: Analyze the relationship between \(\angle1\) and \(\angle4\)

We assume that \(\angle1\) and \(\angle4\) form a linear pair (or \(\angle1\) and \(\angle4\) are supplementary). So \(m\angle1 + m\angle4 = 180^{\circ}\).

Step 2: Calculate \(m\angle4\)

Given \(m\angle1=140^{\circ}\), then \(m\angle4 = 180^{\circ}-m\angle1=180^{\circ}- 140^{\circ} = 40^{\circ}\).
So the statement "If \(m\angle1 = 140^{\circ}\), then \(m\angle4 = 40^{\circ}\)" is True (T).

Question 3

Step 1: Recall the property of vertical angles

If two angles are vertical angles, then they are congruent. We assume that \(\angle2\) and \(\angle4\) are vertical angles.
So \(\angle2\cong\angle4\) (they are congruent).
So the statement "\(\angle2\) and \(\angle4\) are congruent" is True (T).

Question 4

Step 1: Analyze the sum of angles

We know that if \(\angle1\) and \(\angle3\) form a linear pair, \(m\angle1 + m\angle3=180^{\circ}\), and if \(\angle2\) and \(\angle4\) are vertical angles (\(\angle2\cong\angle4\)) and \(\angle1\) and \(\angle2\) form a linear pair (\(m\angle1 + m\angle2 = 180^{\circ}\)), \(\angle3\) and \(\angle4\) form a linear pair (\(m\angle3 + m\angle4=180^{\circ}\)).
Since \(m\angle1 + m\angle3=180^{\circ}\) and \(m\angle2 + m\angle4 = 180^{\circ}\) (because \(\angle2\cong\angle4\) and \(\angle3\) and \(\angle4\) are supplementary, \(\angle1\) and \(\angle2\) are supplementary), then \(m\angle1 + m\angle3=m\angle2 + m\angle4\).
So the statement "\(m\angle3 + m\angle1=m\angle4 + m\angle2\)" is True (T).

Question 5

Step 1: Analyze the relationship between \(\angle3\) and \(\angle4\)

We assume that \(\angle3\) and \(\angle4\) form a linear pair, so \(m\angle3 + m\angle4 = 180^{\circ}\). For \(\angle3\cong\angle4\), we would need \(m\angle3=m\angle4 = 90^{\circ}\), but we are not given that. In general, if they are a linear pair (non - right angles), they are not congruent.
So the statement "\(\angle3\cong\angle4\)" is False (F).

Question 6

Step 1: Analyze the relationship between \(\angle2\) and \(\angle3\)

We assume that \(\angle2\) and \(\angle3\) form a linear pair. By the definition of a linear pair, \(m\angle2 + m\angle3=180^{\circ}\), which can be rewritten as \(m\angle3=180^{\circ}-m\angle2\).
So the statement "\(m\angle3 = 180^{\circ}-m\angle2\)" is True (T).

Answer:

s:

1. T
2. T
3. T
4. T
5. F
6. T