<1 ≅ <3
A Alternate Interior Angles Converse
B Same side interior angles
C Transitive Property
D Given
E Vertical angles are congruent
F Symmetric Property
B Same side interior angles
The statement "<1 ≅ <3" is a statement about angle congruence. To determine which property or theorem can be used to prove or justify this statement, let's examine each option:
A) Alternate Interior Angles Converse: This property states that if two lines are intersected by a transversal and the alternate interior angles are congruent, then the lines are parallel. However, this property does not directly relate to the given statement "<1 ≅ <3".
B) Same Side Interior Angles: This property states that if two lines are intersected by a transversal and the alternate interior angles are supplementary, then the lines are parallel. Again, this property does not directly relate to the given statement "<1 ≅ <3".
C) Transitive Property: This property states that if two statements are separately congruent to a third statement, then they are congruent to each other. The transitive property can be used to prove that if <1 ≅ <2 and <2 ≅ <3, then <1 ≅ <3. However, the given statement "<1 ≅ <3" does not provide any information about other angles that could be used to apply the transitive property.
D) Given: This option suggests that there is some prior information not provided in the question that establishes the congruence of <1 and <3. Without this additional information, we cannot determine if the given statement is justified by the "Given" information alone.
E) Vertical angles are congruent: This property states that when two lines intersect, the vertical angles formed are congruent. Since the question does not mention vertical angles or lines intersecting, this property does not apply to the given statement "<1 ≅ <3".
F) Symmetric Property: This property states that if <1 ≅ <3, then <3 ≅ <1. It involves reversing the order of comparison without altering the congruence. Therefore, if the given statement is true, the symmetric property can be used to also conclude that <3 ≅ <1.
Based on the options provided, the most relevant property to justify the statement "<1 ≅ <3" is the "Symmetric Property" (Option F).
The given statement is <1 ≅ <3. We need to determine which of the options A to F is the correct property or principle that justifies the given statement. Let's go through each option and see which one applies.
A. Alternate Interior Angles Converse: The alternate interior angles converse states that if two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel. Since this property involves parallel lines, it is not applicable to the given statement.
B. Same side interior angles: The same side interior angles refer to the angles that are on the same side of the transversal, one inside each of the parallel lines. This property does not directly relate to the given statement.
C. Transitive Property: The transitive property states that if two things are equal to the same thing, then they are equal to each other. This property is not directly applicable to the given statement.
D. Given: The given option indicates that the statement is true based on a given information. However, we don't know the context or any specific information that has been provided, so this option cannot be applied.
E. Vertical angles are congruent: Vertical angles are the pair of opposite angles formed by two intersecting lines. According to the vertical angles theorem, vertical angles are always congruent. This option matches the given statement, since <1 and <3 are vertical angles and are therefore congruent.
F. Symmetric Property: The symmetric property states that if a = b, then b = a. This property is not directly applicable to the given statement.
Based on the analysis, the correct option is E - Vertical angles are congruent.