Question

Given: r || s and q is a transversal

Answer 1

Answer:

Given: $r || s$ and q is a transversal.

Alternative Interior Angle states that  a pair of angles on the inner side of each of those two lines but on opposite sides of the transversal are equal.

By alternative interior angle;

$\angle 3 \cong \angle 6$

Definition of Congruent angles are angles that have the same degree of measurement.

Therefore, $m\angle 3 = m\angle 6$    [By definition of Congruent]  .....[1]

Linear Pair states that a pair of adjacent angles formed when two lines are intersect.

therefore, $\angle 4$ and $\angle 3$ are a linear pair.   [by definition of linear pair]

Two angles of linear pairs are always supplementary , which means their measure are add up to 180 degree.

By the definition of supplementary angles, $m\angle 4 + m\angle 3 = 180^{\circ}$                 .....[2]

Substitute equation [1] in [2] we get,

$m\angle 4 +m\angle 6 =180^{\circ}$

By the definition of supplementary angles,

$\angle 4$ is supplementary to $\angle 6$   Hence proved!

Answer 2

Answer:

Step-by-step explanation:

Given: r || s and q is a transversal .

To Prove: ∠4 is supplementary to ∠6 .

Proof: It is given that r is parallel to s and q is a transversal, then

∠3 ≅ ∠6 by the (Alternate interior angles).

Therefore, m∠3 = m∠6 by the definition of congruency.

We also know that, by definition, ∠4 and ∠3 are a linear pair, so they are supplementary by the linear pair postulate.

By the definition of supplementary angles, m∠4 + m∠3 = 180°.      (1)

Now, Using substitution,

We can replace m∠3 with m∠6 in equation (1) to get m∠4 + m∠6 = 180°. Therefore, by the definition of supplementary angles, ∠4 is supplementary to ∠6.

Hence proved.