Question
According to the given information, segment JK is parallel to segment HI and points L, N, M, and O all lie on the same line. The measure of ∠LNM 180° by the definition of a straight angle. Because ∠JNL and ∠JNM are adjacent to one another, the Angle Addition Postulate allows the measure of ∠JNL and ∠JNM to equal the measure of ∠LNM. Through the Substitution Property of Equality, the measure of ∠JNL plus the measure of ∠JNM equals 180°. Since ∠JNM and ∠HMN are same-side interior angles, the measure of ∠JNM plus the measure of ∠HMN equals 180°. Using substitution again, the measure of ∠JNL plus the measure of ∠JNM equals the measure of ∠JNM plus the measure of ∠HMN. Finally, the Subtraction Property of Equality allows the measure of ∠JNM to be subtracted from both sides of the equation. The result is that the measure of ∠JNL is the same as the measure of ∠HMN. Because their angle measures are equal, the angles themselves are congruent by the definition of congruency.
Alternate Interior Angles Theorem
Corresponding Angles Theorem
Vertical Angles Theorem
Same-Side Interior Angles Theorem
Alternate Interior Angles Theorem
Corresponding Angles Theorem
Vertical Angles Theorem
Same-Side Interior Angles Theorem
Answers
GPT-4o mini
The argument you provided describes a scenario involving angles formed by a transversal intersecting two parallel lines (segments JK and HI). Here's how the theorems apply:
1. **Alternate Interior Angles Theorem**: This theorem states that if two parallel lines are cut by a transversal, the pairs of alternate interior angles are congruent. In your scenario, angles JNL and HMN are situated in this way, and you concluded they are congruent.
2. **Corresponding Angles Theorem**: This states that corresponding angles formed when a transversal crosses parallel lines are congruent. While you didn't specifically mention this in your description, it's related to the discussion of the angles involved.
3. **Vertical Angles Theorem**: This theorem indicates that the opposite angles formed by two intersecting lines are congruent. While it might not directly apply to your pairs of angles in the problem, it can be another angle relationship used in geometric discussions.
4. **Same-Side Interior Angles Theorem**: This theorem states that when a transversal intersects two parallel lines, the same-side interior angles are supplementary. In your case, angles JNM and HMN apply here, as you noted that they sum to 180°.
Overall, the correct application in your case appears to be the **Alternate Interior Angles Theorem**. You concluded that angles JNL and HMN are congruent based on the parallel lines and the transversal's properties. The measures of the other angles and the properties you've applied (Angle Addition Postulate, Substitution Property of Equality, and Subtraction Property of Equality) also enhance the logic of your argument.
Thus, by invoking the **Alternate Interior Angles Theorem**, you confirm that ∠JNL ≅ ∠HMN.
1. **Alternate Interior Angles Theorem**: This theorem states that if two parallel lines are cut by a transversal, the pairs of alternate interior angles are congruent. In your scenario, angles JNL and HMN are situated in this way, and you concluded they are congruent.
2. **Corresponding Angles Theorem**: This states that corresponding angles formed when a transversal crosses parallel lines are congruent. While you didn't specifically mention this in your description, it's related to the discussion of the angles involved.
3. **Vertical Angles Theorem**: This theorem indicates that the opposite angles formed by two intersecting lines are congruent. While it might not directly apply to your pairs of angles in the problem, it can be another angle relationship used in geometric discussions.
4. **Same-Side Interior Angles Theorem**: This theorem states that when a transversal intersects two parallel lines, the same-side interior angles are supplementary. In your case, angles JNM and HMN apply here, as you noted that they sum to 180°.
Overall, the correct application in your case appears to be the **Alternate Interior Angles Theorem**. You concluded that angles JNL and HMN are congruent based on the parallel lines and the transversal's properties. The measures of the other angles and the properties you've applied (Angle Addition Postulate, Substitution Property of Equality, and Subtraction Property of Equality) also enhance the logic of your argument.
Thus, by invoking the **Alternate Interior Angles Theorem**, you confirm that ∠JNL ≅ ∠HMN.