051i - Co-interior Angles

Lesson 51 — Co-interior Angles

Strand: Measurement | Descriptor: AC9M7M04 | Duration: 45 minutes

Learning Intentions

• To identify co-interior angle relationships formed when parallel lines are crossed by a transversal.
• To use the co-interior property to find unknown angles and justify the reasoning.

Success Criteria

I can:

1. Recognise co-interior angles by their C or U shape.
2. State that co-interior angles on parallel lines are supplementary.
3. Derive the co-interior property from the alternate angle property.
4. Choose correctly between corresponding, alternate and co-interior relationships.

Warmup

(6 minutes — retrieval from Lesson 50, mini whiteboards)

Two parallel lines are crossed by a transversal. One angle is .

1. Find the corresponding angle. Name the reason.
2. Find the alternate angle. Name the reason.
3. Find the vertically opposite angle.
4. Find the angle on a straight line with it.
5. How many different angle sizes appear among the eight angles?

Answers: 1. (corresponding angles); 2. (alternate angles); 3. (vertically opposite); 4. (angles on a straight line); 5. Two — and .

Bridging question: In Q4 you found . Is there an angle at the other intersection that is also and paired with the in a meaningful way? That pairing is today’s topic.

Activities

Activity 1 — Explicit Instruction: Co-interior Angles (14 min)

Definition. Co-interior angles lie on the same side of the transversal and between the two parallel lines. They are sometimes called allied or supplementary angles.

The C or U shape. Trace the letter C (or U) over the diagram: the two angles inside the curve are co-interior. As with F and Z, the letter may be rotated or reflected.

The rule:

When a transversal crosses parallel lines, co-interior angles sum to .

Note carefully: this is the one relationship of the three where the angles are not equal.

Deriving the rule — do not assert it. Using the labelling from Lesson 50 (angles – upper, – lower):

So co-interior follows from alternate, which followed from corresponding. All three rules trace back to one idea.

The complete summary table — display for the rest of the unit:

Relationship	Shape	Position	Rule
Corresponding	F	same side of transversal, matching positions	equal
Alternate	Z	opposite sides, between the lines	equal
Co-interior	C or U	same side, between the lines	sum to
Vertically opposite	X	at one intersection, opposite each other	equal

I do. Given a co-interior angle of , find its partner.

We do: Find the co-interior partner of , , .

(Answers: ; ; .)

Discussion of the case: when the transversal is perpendicular, co-interior angles are both — equal and supplementary. This is the only case where all three relationships give the same value.

Activity 2 — Choosing the Right Relationship (12 min)

The main difficulty is now selection, not calculation.

The three-question decision procedure — teach explicitly:

1. Are the two angles between the parallel lines? If no → corresponding.
2. If yes, are they on the same side of the transversal? If yes → co-interior (sum ).
3. If no (opposite sides) → alternate (equal).

You do: For each pair described, name the relationship and find the unknown.

1. Same side of the transversal, both between the lines; one is .
2. Opposite sides, both between the lines; one is .
3. Same position at each intersection; one is .
4. Same side, both between the lines; one is .
5. Opposite sides, both between the lines; one is .
6. Matching positions, one above each parallel line and both left of the transversal; one is .

(Answers: 1. co-interior, ; 2. alternate, ; 3. corresponding, ; 4. co-interior, ; 5. alternate, ; 6. corresponding, .)

Common trap to name. Students who have just learnt “corresponding and alternate are equal” often apply equal to co-interior as well. The mnemonic: C is for co-interior and for “complete to “.

Activity 3 — Inquiry: the Zigzag Path (10 min)

Pairs.

Two parallel walls run north–south, m apart. A path zigzags between them, striking the west wall at and reflecting to the east wall.

1. Draw the situation and mark the angle.
2. Find the angle the path makes with the east wall.
3. Find the co-interior angle at the east wall.
4. If the path strikes at each time, what angle does the zigzag turn through at each wall?

Socratic scaffolding:

Prompt	Purpose
Understand: what plays the role of the transversal?	The path segment crossing between the two walls.
The walls are parallel. Which relationship links the two positions?	Alternate angles — opposite sides of the path, between the walls.
So what is the angle at the east wall?	Also .
Now the co-interior angle at that wall.	.
What angle does the path turn through at a wall?	The path arrives at and leaves at on the other side, so it turns through .
Looking back — check the picture.	An turn at each bounce gives a narrow zigzag, consistent with a shallow strike. ✓

Answers: 2. (alternate angles); 3. (co-interior); 4. .

Extension: what strike angle would make the path turn through exactly at each wall? (Solving gives .)

Checks for Understanding

(5 minutes — exit ticket)

1. Find the co-interior partner of an angle of .
2. Two parallel lines are crossed by a transversal. Co-interior angles are and . Find and both angles.
3. Name the relationship: two angles on the same side of the transversal and between the parallel lines.
4. Reasoning. Explain why co-interior angles sum to , using alternate angles.
5. A student writes “co-interior angles are equal, so .” Identify and correct the error.

Answers: 1. ; 2. , so ; angles and ; 3. Co-interior angles; 4. One co-interior angle equals the alternate angle at the other intersection, and that alternate angle sits on a straight line with the second co-interior angle — so together they total ; 5. Co-interior angles are supplementary, not equal, so .

Common Misconceptions

Misconception	How to pre-empt it
Treating co-interior angles as equal.	The “C for complete to ” mnemonic, reinforced every time the relationship is used.
Confusing co-interior with alternate.	Both sit between the lines — the deciding question is which side of the transversal.
Applying the rule when the lines are not parallel.	Check for arrowhead markings first, every time.
Assuming the C shape must open a particular way.	Rotate the diagram and re-identify the same pair.
Adding all three relationships to indiscriminately.	Only co-interior sums to ; corresponding and alternate are equal.
Omitting the reason in written work.	Reasons must name the relationship and the parallel condition.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Two parallel lines are crossed by a transversal. Co-interior angles are and . Find .

Answer

The angles are and .

E2 (AMC Junior style). Two parallel lines are crossed by a transversal. One co-interior angle is four times the other. Find both.

Answer

The angles are and .

E3 (Challenge). In a parallelogram, why must adjacent angles sum to ?

Answer

A parallelogram has two pairs of parallel sides. Any side acts as a transversal crossing the two parallel sides it joins, so adjacent angles are co-interior — and therefore supplementary. (This explains the parallelogram angle property met in Lesson 42.)

E4 (Challenge). Two parallel lines are crossed by a transversal so that one co-interior angle is more than three times the other. Find both.

Answer

The angles are and .

E5 (Reasoning challenge). Explain why, if co-interior angles sum to , the two lines must be parallel.

Answer

If the lines were not parallel they would meet, forming a triangle with the transversal. The two co-interior angles would then be two angles of that triangle, and their sum would be less than (leaving room for the third angle). Since they total exactly , no such triangle can exist, so the lines never meet — they are parallel.

Homework

1. Find the co-interior partner of: (a) (b) (c) (d) (e) .
2. Name the relationship and find the unknown: (a) Same side of the transversal, both between the lines; one is . (b) Opposite sides, both between the lines; one is . (c) Matching positions at each intersection; one is . (d) Same side, both between the lines; one is .
3. Co-interior angles are and . Find and both angles.
4. Co-interior angles are and . Find and both angles.
5. Two parallel lines are crossed by a transversal, and one angle is . State all eight angle sizes and name one relationship for each distinct value.
6. Reasoning. Explain the difference between alternate and co-interior angles, referring to the transversal.
7. Reasoning. Explain why co-interior angles are both when the transversal is perpendicular to the parallel lines.
8. Challenge. In a parallelogram, one angle is . Use co-interior angles to find the other three.
9. Challenge. Co-interior angles are in the ratio . Find both.

Answers: Q1 — (a) (b) (c) (d) (e) . Q2 — (a) co-interior, (b) alternate, (c) corresponding, (d) co-interior, . Q3 — , so ; angles and . Q4 — , so ; angles and . Q5 — four of and four of . Q7 — a perpendicular transversal makes every angle , and ✓. Q8 — . Q9 — parts total , so one part is ; the angles are and .