Respuesta :
Answer:
Given: [tex]m\angle 1 = m\angle 3[/tex] and [tex]m\angle 2 = m\angle 3[/tex]
To prove that:
[tex]l || m[/tex]
1. [tex]m\angle 1 = m\angle 3[/tex] [Given]
[tex]m\angle 2 = m\angle 3[/tex]
Substitution property of equality says that:
If x = y, then x can be substituted in y, or y can be substituted in x.
2 [tex]m\angle 1 = m\angle 2[/tex] [ By Substitution Property]
Alternate interior angles states that when two lines are crossed by transversal , a pair of angles on the inner sides of each of these two lines on the opposite sides of the transversal line.
3. [tex]\angle 1[/tex] and [tex]\angle 2[/tex] are alternate interior angles [By definition Alternate interior angle].
Alternating interior angles theorem states that if two parallel lines are intersected by third lines, then the angles in the inner sides of the parallel lines on the opposite sides of the transversal are equal.
4. [tex]l || m[/tex] ; then the lines are parallel [By Alternate interior angles theorem]
Correct match is as follows:
1. [tex]m\angle 1 = m\angle 3[/tex] [Given]
[tex]m\angle 2 = m\angle 3[/tex]
2. [tex]m\angle 1 = m\angle 2[/tex] [Substitution]
3. [tex]\angle 1[/tex] and [tex]\angle 2[/tex] are alternate interior angle [By definition of alternate interior angles ]
4. [tex]l || m[/tex] the lines are parallel [If alternate interior angles are equal]
The given list of statements and reasons why they are true can be
presented in a two column proof.
The reasons and statements are as follows;
Statement [tex]{}[/tex] Reason:
1. m∠1 = m∠3 [tex]{}[/tex] Given
m∠2 = m∠3 [tex]{}[/tex] Given
2. m∠1 = m∠2 [tex]{}[/tex] Substitution property of equality
3. ∠1 and ∠2 are alternate interior angles (Definition of alternate interior angles)
4. l║m [tex]{}[/tex] If Alternate interior angles are equal then the lines are parallel.
Reasons:
The proof can be presented as follows;
- m∠1 = m∠3, and m∠2 = m∠3, given
- By substituting m∠3 with m∠1, we get; m∠1 = m∠2
- ∠1 and ∠2 are alternate interior angles based on their relative position relative to the common transversal and the two parallel lines
- The alternate interior angles m∠1 and m∠2 are equal, therefore, by alternate interior angles theorem, we have l║m.
Therefore, the two column proof is presented as follows;
Statement [tex]{}[/tex] Reason:
1. m∠1 = m∠3 [tex]{}[/tex] Given
m∠2 = m∠3 [tex]{}[/tex] Given
2. m∠1 = m∠2 [tex]{}[/tex] Substitution property of equality
3. ∠1 and ∠2 are alternate interior angles (Definition of alternate interior angles)
4. l║m [tex]{}[/tex] If Alternate interior angles are equal then the lines are parallel.
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