8.G.A.5
8th Grade
Geometry
Angle Relationships
Use informal arguments to establish facts about angle sums, exterior angles, and angles formed when parallel lines are cut by a transversal.
How to explain it
The anchor students hold onto: Congruent pairs: corresponding · alternate interior · alternate exterior · vertical. Supplementary pairs: co-interior · linear pair (sum = 180°).
Angle pair facts extend directly to #96 Pythagorean Theorem and to the angle-angle (AA) similarity criterion for triangles later in 8th grade.
Worked examples
Example 1
Corresponding angles
Find ∠5; ℓ₁ ∥ ℓ₂, ∠1 = 55°.
Step 1Corresponding pairs: same position at each intersection.
Step 2∠1 and ∠5 are both upper-left at their intersections.
Answer∠5 = 55° (congruent).
Example 2
Co-interior angles
Find ∠6; ℓ₁ ∥ ℓ₂, ∠3 = 112°.
Step 1∠3 and ∠6 are co-interior: same side of t, between the parallel lines.
Step 2∠3 + ∠6 = 180°. 112° + ∠6 = 180°.
Answer∠6 = 68° (supplementary).
Example 3
Vertical angles
At P, find ∠3 if ∠1 = 55°.
Step 1Vertical pairs form at each intersection (opposite rays).
Step 2∠1 and ∠3 are vertical at P.
Answer∠3 = 55° (congruent).
Common mistakes
What students write
Calling co-interior pairs congruent.
The fix
Co-interior pairs are SUPPLEMENTARY — they sum to 180°, not share a measure.
Try this
A student solved: Lines ℓ₁ ∥ ℓ₂ cut by t. ∠4 = 97°. Find ∠5. Student wrote: '∠4 and ∠5 are alternate interior angles. ∠5 = ∠4 = 97°.' Describe the student's error. Find the correct ∠5.
What students write
Naming pairs without confirming the lines are parallel.
The fix
The named relationships only hold when ℓ₁ ∥ ℓ₂. Always verify parallelism (tick marks or stated) first.
Try this
A student found ∠5 when ∠4 = 108°. Student wrote: '∠4 and ∠5 are corresponding because they are at different intersections. ∠5 = ∠4 = 108°.' Identify the error. Find the correct ∠5.
Teacher tip
Head off the two predictable errors before they happen. First: Co-interior pairs are SUPPLEMENTARY — they sum to 180°, not share a measure. Second: The named relationships only hold when ℓ₁ ∥ ℓ₂. Always verify parallelism (tick marks or stated) first.