Is there a relation between and ? You could try to find the relationship by taking different values for and see what is. Once you find a relation, try to justify it or prove that this relation holds always.

Understand the Question When two parallel lines are intersected by a transversal, the interior angles on the same side of the transversal (co-interior angles) are supplementary. This means their measures add up to . We can verify this with a numerical example and formally prove that using vertically opposite angles, corresponding angles, and linear pairs. Step 1 · Find the Relationship and Test with a Value Lines and are parallel (), and line is the transversal. Angles and are consecutive interior angles on the same side of transversal , so they are supplementary: Taking : Step 2 · Justify and Prove the Relation To prove that holds for any value when : 1. and are vertically opposite angles: 2. and are corresponding angles for : From (1) and (2): 3. and form a linear pair on line : Substitute into the equation: Hence, the relation always holds. Answer > (the angles are supplementary) Common Mistakes Assuming Equal Measures: Mistaking interior angles on the same side (co-interior angles) for alternate interior angles and incorrectly equating them as instead of . Overlooking Parallel Condition: Forgetting that the supplementary relation holds strictly when lines and are parallel.

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Question 18

Is there a relation between $\angle 3$ and $\angle 6$ ? You could try to find the relationship by taking different values for $\angle 3$ and see what $\angle 6$ is. Once you find a relation, try to justify it or prove that this relation holds always.

Solution

Understand the Question

When two parallel lines are intersected by a transversal, the interior angles on the same side of the transversal (co-interior angles) are supplementary.
This means their measures add up to $180^\circ$ .
We can verify this with a numerical example and formally prove that $\angle 3 + \angle 6 = 180^\circ$ using vertically opposite angles, corresponding angles, and linear pairs.

Step 1 · Find the Relationship and Test with a Value

Lines $l$ and $m$ are parallel ( $l \parallel m$ ), and line $t$ is the transversal. Diagram 1

Angles $\angle 3$ and $\angle 6$ are consecutive interior angles on the same side of transversal $t$ , so they are supplementary: $\angle 3 + \angle 6 = 180^\circ$

Taking $\angle 3 = 50^\circ$ : $50^\circ + \angle 6 = 180^\circ$

\begin{aligned} \angle 6 &= 180^\circ - 50^\circ \\ &= 130^\circ \end{aligned}

Step 2 · Justify and Prove the Relation

To prove that $\angle 3 + \angle 6 = 180^\circ$ holds for any value when $l \parallel m$ :

$\angle 1$ and $\angle 3$ are vertically opposite angles: $\angle 1 = \angle 3$
$\angle 1$ and $\angle 5$ are corresponding angles for $l \parallel m$ : $\angle 1 = \angle 5$

From (1) and (2): $\angle 3 = \angle 5$

$\angle 5$ and $\angle 6$ form a linear pair on line $m$ : $\angle 5 + \angle 6 = 180^\circ$

Substitute $\angle 5 = \angle 3$ into the equation: $\angle 3 + \angle 6 = 180^\circ$

Hence, the relation $\angle 3 + \angle 6 = 180^\circ$ always holds.

Answer

$\angle 3 + \angle 6 = 180^\circ$ (the angles are supplementary)

Common Mistakes

Assuming Equal Measures: Mistaking interior angles on the same side (co-interior angles) for alternate interior angles and incorrectly equating them as $\angle 3 = \angle 6$ instead of $\angle 3 + \angle 6 = 180^\circ$ .
Overlooking Parallel Condition: Forgetting that the supplementary relation $\angle 3 + \angle 6 = 180^\circ$ holds strictly when lines $l$ and $m$ are parallel.

Question 18

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