Question

∠f and ∠h are supplementary and ∠f and ∠g are supplementary. complete the proof that ∠g ≅ ∠h. 1 ∠f and ∠h are supplementary 2 ∠f and ∠g are supplementary 3 ( m angle f + m angle h = 180 ^ { circ } ) 4 ( m angle f + m angle g = 180 ^ { circ } ) 5 ( m angle f + m angle h = m angle f + m angle g ) 6 ( m angle g = m angle h ) 7 ∠g ≅ ∠h

Explanation:

Step1: Given

Statement 1 and 2 are given in the problem.

Step2: Definition of supplementary angles

Since \(\angle F\) and \(\angle H\) are supplementary, \(m\angle F + m\angle H=180^{\circ}\) (by definition of supplementary angles: two angles are supplementary if the sum of their measures is \(180^{\circ}\)). Similarly, for \(\angle F\) and \(\angle G\), \(m\angle F + m\angle G = 180^{\circ}\) (by definition of supplementary angles).

Step3: Transitive property of equality

If \(a = b\) and \(a=c\), then \(b = c\). Here, \(a=m\angle F + 180^{\circ}\), \(b=m\angle F + m\angle H\) and \(c=m\angle F + m\angle G\). So, \(m\angle F + m\angle H=m\angle F + m\angle G\) (transitive property of equality: if two expressions are equal to the same value, then they are equal to each other).

Step4: Subtraction property of equality

Subtract \(m\angle F\) from both sides of the equation \(m\angle F + m\angle H=m\angle F + m\angle G\). We get \(m\angle H=m\angle G\) (subtraction property of equality: if \(a + b=a + c\), then \(b = c\)).

Step5: Definition of congruent angles

If \(m\angle G=m\angle H\), then \(\angle G\cong\angle H\) (by definition of congruent angles: two angles are congruent if their measures are equal).

Answer:

1. Given
2. Given
3. Definition of supplementary angles
4. Definition of supplementary angles
5. Transitive property of equality
6. Subtraction property of equality
7. Definition of congruent angles