Question

#3.) in the diagram below l || m.
fill in the chart below:

angles	angle pair name	relationship
∠5 and ______	vertical angles
∠7 and ______	alternate interior angles
∠10 and ∠11
∠8 and ∠16
∠15 and ______	same side exterior angles

#4.) in the diagram below $overline{ab}paralleloverline{cd}$.
fill in the chart below:

angle pair name	angles	relationship
same side exterior angles	∠bge and ∠______
alternate interior angles	∠agh and ∠______
alternate exterior angles	∠chf and ∠______
corresponding angles	∠chg and ∠______

Explanation:

Step1: Recall angle - pair relationships

When two parallel lines are cut by a transversal, different angle - pair relationships exist. Vertical angles are opposite each other at an intersection of two lines. Alternate interior angles are between the two parallel lines and on opposite sides of the transversal. Same - side interior angles are between the two parallel lines and on the same side of the transversal. Same - side exterior angles are outside the two parallel lines and on the same side of the transversal. Alternate exterior angles are outside the two parallel lines and on opposite sides of the transversal. Corresponding angles are in the same relative position with respect to the parallel lines and the transversal.

Step2: Analyze ∠2 and ∠12

∠2 and ∠12 are corresponding angles. The relationship is that corresponding angles are congruent when \(l\parallel m\).

Step3: Find vertical angle for ∠5

The vertical angle for ∠5 is ∠7. Vertical angles are congruent.

Step4: Find alternate interior angle for ∠7

The alternate interior angle for ∠7 is ∠10. Alternate interior angles are congruent when \(l\parallel m\).

Step5: Analyze ∠10 and ∠11

∠10 and ∠11 are adjacent angles. They share a common side and a common vertex.

Step6: Analyze ∠8 and ∠16

∠8 and ∠16 are corresponding angles. They are congruent when \(l\parallel m\).

Step7: Find same - side exterior angle for ∠15

The same - side exterior angle for ∠15 is ∠1.

Step8: For #4, find same - side interior angle for ∠BGH

The same - side interior angle for ∠BGH is ∠GHD. Same - side interior angles are supplementary when \(AB\parallel CD\).

Step9: Find same - side exterior angle for ∠BGE

The same - side exterior angle for ∠BGE is ∠DHF.

Step10: Find alternate interior angle for ∠AGH

The alternate interior angle for ∠AGH is ∠GHD.

Step11: Find alternate exterior angle for ∠CHF

The alternate exterior angle for ∠CHF is ∠AGE.

Step12: Find corresponding angle for ∠CHG

The corresponding angle for ∠CHG is ∠AGH.

Answer:

Angles	Angle Pair Name	Relationship
∠5 and ∠7	Vertical Angles	Congruent
∠7 and ∠10	Alternate Interior Angles	Congruent
∠10 and ∠11	Adjacent Angles	Share common side and vertex
∠8 and ∠16	Corresponding Angles	Congruent
∠15 and ∠1	Same Side Exterior Angles	Supplementary (when \(l\parallel m\))
Angle Pair Name	Angles	Relationship
Same Side Interior Angles	∠BGH and ∠GHD	Supplementary
Same Side Exterior Angles	∠BGE and ∠DHF	Supplementary
Alternate Interior Angles	∠AGH and ∠GHD	Congruent
Alternate Exterior Angles	∠CHF and ∠AGE	Congruent
Corresponding Angles	∠CHG and ∠AGH	Congruent