Question

Match the reasons with the statements. Complete the proof for Theorem 3-16 above, with the information given. Given: m1 = m2 Prove: l | | m Plan: Same as Theorem 3-15 1. m∠1 = m∠2 If corresponding angles are equal, then lines are parallel. 2. m∠1 = m∠3 Given 3. m∠2 = m∠3 Substitution 4. l||m Vertical angles are equal.

Answer 1

Answer:

Given: m1 = m2

If corresponding angles equal, lines are parallel: m2 = m3

Vertical angles are equal: l||m

Substitution: m1 = m3

Answer 2

The statements and reasons of the complete proof showing that lines l and m are parallel to each other are:

Statement 1: $\angle 1 \cong \angle 2$

Reason: Given

Statement 2: $\angle 1 \cong \angle 3$

Reason: Vertical angles are equal

Statement 3: $\angle 2 \cong \angle 3$

Reason: Substitution

Statement 4: $l \parallel m$

Reason: corresponding angles theorem

Recall:

• When a transversal cuts across two lines, the two lines can be proven to be parallel if their corresponding angles are congruent to each other.

Considering the image attached below, we are given that, $\angle 1 \cong \angle 2$.

The complete proof showing the reasons and statements that proves that $l \parallel m$ is shown below:

Statement 1: $\angle 1 \cong \angle 2$

Reason: Given

Statement 2: $\angle 1 \cong \angle 3$

Reason: Vertical angles are equal (<1 and <3 are directly opposite each other, they are vertical angles).

Statement 3: $\angle 2 \cong \angle 3$

Reason: Substitution (<1 = <2, <1 = <3, therefore, by substitution, <2 = <3)

Statement 4: $l \parallel m$

Reason: corresponding angles theorem (we have proved that the corresponding angles, <2 and <3 are congruent or equal, therefore, it implies that lines l and m are parallel to each other).

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