Consider parallel lines cut by a transversal. Parallel lines q and s are cut by transversal r. On line q where it intersects line r, 4 angles are created. Labeled clockwise, from uppercase left: angle 1, angle 2, angle 4, angle 3. On line s where it intersects line r, 4 angles are created. Labeled clockwise, from uppercase left: angle 5, angle 6, angle 8, angle 7. Explain which theorems, definitions, or combinations of both can be used to prove that alternate exterior angles are congruent

The relationship between angles formed between parallel lines includes; alternate exterior and interior, corresponding, vertical, and same side interior and exterior angles

The theorems, definitions, that can be used to prove that alternate exterior angles are congruent are;

1) Corresponding angles postulate

2) Vertical angles theorem

3) Transitive property of congruency

The explanation and reasons on how they are applied as proof are as follows:

Question: Please find attached the diagram representing the question

Required:

The combination of theorems, definitions, or combinations of both that can be used to prove that alternate exterior angles are congruent

Solution:

The alternate exterior angles are the angle formed by the intersection of two parallel lines and a common transversal that are located on the outer side of the parallel lines and on opposite sides of the transversal

The alternate exterior angles are;

∠7 and ∠2

∠8 and ∠1

The theorems that prove that ∠7 and ∠2 are congruent are presented in the following two column proof;

[tex]\begin{array}{lcr} \mathbf{Statements}&&\mathbf{Reason}\\1. \ q \parallel s&&Given\\2. \ \angle 6 \cong \angle 2 &&Corresponding \ angles \ \mathbf{postulate}\\3. \ \angle 6 \cong \angle 7&&Vertical \ angles \ \mathbf{theorem}\\4. \ \angle 7 \cong \angle 2&&By \ the \ transitive \ \mathbf{property} \ of \ congruency\end{array}[/tex]

The same theorems, postulate, and property, can be used to prove that ∠8 and ∠1 are congruent

The descriptions of the postulates, theorem and properties that prove alternate exterior angles are congruent are as follows:

The corresponding angles postulate states that angles formed on the corresponding sides of two parallel having a common transversal are equal

The vertical angles theorem states that the opposite angles formed when two lines intersect are always equal

The transitive property of congruency proves that if two angles are congruent, and if one of the angles is also congruent to a third angle, then the first angle is also congruent to the third angle

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