Alternate Exterior Angles
Alternate exterior angles are pairs of angles that lie on opposite sides of a transversal and outside the two lines it intersects. When the two lines are parallel, these angle pairs are always equal.
This property is a direct consequence of the corresponding angles axiom and the vertically opposite angles theorem. In Class 9 Geometry, it is used frequently in formal proofs involving parallel lines.
Understanding alternate exterior angles complements the study of alternate interior angles, corresponding angles, and co-interior angles — all formed when a transversal cuts two lines.
What is Alternate Exterior Angles?
Definition: When a transversal intersects two lines, alternate exterior angles are the pairs of angles that are:
- On opposite sides (alternate sides) of the transversal, AND
- Outside (exterior to) the two lines.
Theorem (Alternate Exterior Angles):
If two parallel lines are cut by a transversal, each pair of alternate exterior angles is equal.
Identifying the pairs:
- Let lines l and m be cut by transversal t.
- Label the angles at line l (top intersection) as ∠1, ∠2, ∠3, ∠4 and at line m (bottom intersection) as ∠5, ∠6, ∠7, ∠8.
- Alternate exterior angle pairs: (∠1, ∠7) and (∠2, ∠8).
- Compare with alternate interior angles: (∠3, ∠5) and (∠4, ∠6).
Converse: If a transversal cuts two lines such that a pair of alternate exterior angles is equal, then the two lines are parallel.
Alternate Exterior Angles Formula
Key Results:
1. Alternate Exterior Angles Theorem:
If l ∥ m, then ∠1 = ∠7 and ∠2 = ∠8
2. Converse:
If ∠1 = ∠7 (or ∠2 = ∠8), then l ∥ m
3. Relationship with other angle pairs:
- Alternate exterior angles = Corresponding angles (via vertically opposite angles).
- ∠1 = ∠5 (corresponding) and ∠5 = ∠7 (vertically opposite) ⇒ ∠1 = ∠7.
- Similarly, ∠2 = ∠6 (corresponding) and ∠6 = ∠8 (vertically opposite) ⇒ ∠2 = ∠8.
Derivation and Proof
Proof: Alternate Exterior Angles are Equal when Lines are Parallel
Given: Lines l ∥ m, cut by transversal t. ∠1 and ∠7 are alternate exterior angles.
To prove: ∠1 = ∠7
Proof:
- ∠1 = ∠5 (Corresponding angles; l ∥ m, cut by t) …(i)
- ∠5 = ∠7 (Vertically opposite angles at the intersection with m) …(ii)
- From (i) and (ii): ∠1 = ∠7 (Transitive property of equality)
Similarly, for the other pair:
- ∠2 = ∠6 (Corresponding angles) …(iii)
- ∠6 = ∠8 (Vertically opposite angles) …(iv)
- From (iii) and (iv): ∠2 = ∠8
Hence proved: Both pairs of alternate exterior angles are equal when the lines are parallel. ■
Proof of the Converse:
Given: ∠1 = ∠7
To prove: l ∥ m
- ∠7 = ∠5 (Vertically opposite angles) …(i)
- ∠1 = ∠7 (Given) …(ii)
- From (i) and (ii): ∠1 = ∠5
- Since ∠1 and ∠5 are corresponding angles and they are equal, by the converse of the corresponding angles axiom, l ∥ m.
Types and Properties
Angle Pairs Formed by a Transversal:
When a transversal cuts two lines, 8 angles are formed (4 at each intersection). The key angle pairs are:
- Same position at each intersection (both above or both below, both left or both right).
- Pairs: (∠1, ∠5), (∠2, ∠6), (∠3, ∠7), (∠4, ∠8).
- Equal when lines are parallel.
- Opposite sides of transversal, between the lines.
- Pairs: (∠3, ∠5), (∠4, ∠6).
- Equal when lines are parallel.
3. Alternate Exterior Angles
- Opposite sides of transversal, outside the lines.
- Pairs: (∠1, ∠7), (∠2, ∠8).
- Equal when lines are parallel.
4. Co-interior (Same-side Interior) Angles
- Same side of transversal, between the lines.
- Pairs: (∠3, ∠6), (∠4, ∠5).
- Supplementary (sum = 180°) when lines are parallel.
5. Co-exterior (Same-side Exterior) Angles
- Same side of transversal, outside the lines.
- Pairs: (∠1, ∠8), (∠2, ∠7).
- Supplementary (sum = 180°) when lines are parallel.
Solved Examples
Example 1: Example 1: Finding alternate exterior angles
Problem: Lines PQ ∥ RS are cut by transversal LM. If ∠PLM = 125°, find the alternate exterior angle at line RS.
Solution:
Given:
- PQ ∥ RS, transversal LM
- ∠PLM = 125° (exterior angle at line PQ)
The alternate exterior angle is at line RS on the opposite side of the transversal.
- By the alternate exterior angles theorem: alternate exterior angle = 125°
Answer: The alternate exterior angle is 125°.
Example 2: Example 2: Using alternate exterior angles to find x
Problem: Two parallel lines are cut by a transversal. One alternate exterior angle is (3x + 10)° and the other is (5x − 30)°. Find x.
Solution:
Since the lines are parallel, alternate exterior angles are equal:
- 3x + 10 = 5x − 30
- 10 + 30 = 5x − 3x
- 40 = 2x
- x = 20
The angles are:
- 3(20) + 10 = 70°
- 5(20) − 30 = 70° ✔
Answer: x = 20; each angle is 70°.
Example 3: Example 3: Proving lines are parallel
Problem: A transversal cuts two lines AB and CD. The alternate exterior angles formed are 115° each. Are the lines parallel?
Solution:
Given:
- Alternate exterior angles = 115° each (equal)
By the converse of the alternate exterior angles theorem:
- If a pair of alternate exterior angles is equal, the lines are parallel.
- Since both angles are 115° (equal), the lines AB and CD are parallel.
Answer: Yes, AB ∥ CD.
Example 4: Example 4: Lines not parallel
Problem: A transversal cuts two lines. The alternate exterior angles are 105° and 110°. Are the lines parallel?
Solution:
Given:
- Alternate exterior angles: 105° and 110°
Check:
- 105° ≠ 110°
- Since the alternate exterior angles are NOT equal, the lines are not parallel.
Answer: No, the lines are not parallel.
Example 5: Example 5: Finding all 8 angles
Problem: Lines l ∥ m are cut by transversal t. One of the angles is 62°. Find all 8 angles.
Solution:
Given: ∠1 = 62° (at the intersection with line l, above-left of transversal)
At line l:
- ∠1 = 62°
- ∠2 = 180° − 62° = 118° (linear pair)
- ∠3 = 118° (vertically opposite to ∠2)
- ∠4 = 62° (vertically opposite to ∠1)
At line m (using corresponding angles, l ∥ m):
- ∠5 = ∠1 = 62° (corresponding)
- ∠6 = ∠2 = 118° (corresponding)
- ∠7 = ∠3 = 118° (corresponding)
- ∠8 = ∠4 = 62° (corresponding)
Verify alternate exterior angles: ∠1 = ∠7? 62 ≠ 118. Wait — let us re-check the labelling.
Standard labelling: ∠1 (above-left at l), ∠2 (above-right at l), ∠3 (below-left at l), ∠4 (below-right at l); ∠5 (above-left at m), ∠6 (above-right at m), ∠7 (below-left at m), ∠8 (below-right at m).
- Alternate exterior pairs: (∠1, ∠8) and (∠2, ∠7)
- ∠1 = 62°, ∠8 = 62° ✔
- ∠2 = 118°, ∠7 = 118° ✔
Answer: The 8 angles are 62°, 118°, 118°, 62°, 62°, 118°, 118°, 62°.
Example 6: Example 6: Combined angle relationships
Problem: In the figure, AB ∥ CD, transversal PQ meets AB at E and CD at F. ∠PEB = 72°. Find ∠EFD.
Solution:
Given:
- AB ∥ CD, transversal PQ
- ∠PEB = 72° (exterior angle at E, on the right side of PQ, above AB)
∠PEB and ∠EFD are alternate exterior angles (on opposite sides of PQ, both exterior to lines AB and CD).
- ∠EFD = ∠PEB = 72°
Answer: ∠EFD = 72°.
Example 7: Example 7: Three parallel lines
Problem: Lines l, m, and n are parallel. A transversal cuts l at A, m at B, and n at C. If the exterior angle at A (above l) on the left of the transversal is 130°, find the exterior angle at C (below n) on the right of the transversal.
Solution:
Given: l ∥ m ∥ n; exterior angle at A = 130°
The angle at A (above l, left) and the angle at C (below n, right) are alternate exterior angles with respect to lines l and n.
- Since l ∥ n, alternate exterior angles are equal.
- Angle at C = 130°
Answer: The angle at C is 130°.
Example 8: Example 8: Using supplementary co-exterior angles
Problem: AB ∥ CD, transversal EF. If the exterior angle at AB on the left is 75°, find the exterior angle at CD on the same side (left) of the transversal.
Solution:
Given: AB ∥ CD; exterior angle on left at AB = 75°
These two angles (same-side exterior, also called co-exterior) are supplementary when lines are parallel.
- 75 + x = 180
- x = 180 − 75 = 105°
Answer: The co-exterior angle is 105°.
Example 9: Example 9: Application in a geometric figure
Problem: In a trapezium ABCD, AB ∥ DC. The diagonal AC acts as a transversal. If ∠BAC = 50° and ∠ACD = 50°, verify that these are alternate interior angles and confirm AB ∥ DC.
Solution:
Given:
- ∠BAC = 50° (between lines AB and DC, on the AB side)
- ∠ACD = 50° (between lines AB and DC, on the DC side)
These are alternate interior angles with respect to lines AB and DC cut by transversal AC.
- Since ∠BAC = ∠ACD = 50°, by the converse of the alternate interior angles theorem, AB ∥ DC ✔
Answer: The equality of these alternate angles confirms AB ∥ DC.
Example 10: Example 10: Mixed angle problem
Problem: PQ ∥ RS, transversal AB. ∠PAB = (4x + 5)° and the co-exterior angle ∠SBA = (6x − 25)°. Find x and both angles.
Solution:
∠PAB and ∠SBA are co-exterior (same-side exterior) angles.
Co-exterior angles are supplementary when lines are parallel:
- (4x + 5) + (6x − 25) = 180
- 10x − 20 = 180
- 10x = 200
- x = 20
The angles are:
- ∠PAB = 4(20) + 5 = 85°
- ∠SBA = 6(20) − 25 = 95°
Verification: 85 + 95 = 180 ✔ (supplementary)
Answer: x = 20; the angles are 85° and 95°.
Real-World Applications
Applications of Alternate Exterior Angles:
- Proving lines parallel: The converse of the alternate exterior angles theorem is a standard method for establishing that two lines are parallel in geometric proofs.
- Architecture and construction: Parallel beams in buildings and bridges create alternate exterior angles with cross-braces. Verifying these angles ensures structural alignment.
- Road and railway design: Parallel lanes cut by diagonal roads form alternate exterior angles. Traffic engineers use these to design consistent turning angles.
- Optical reflections: Parallel mirrors create alternate exterior angles with light paths, used in periscope and laser alignment design.
- Map reading and surveying: Parallel boundary lines cut by survey lines create predictable angle relationships for accurate mapping.
- Proof technique: Alternate exterior angles are often easier to identify visually than alternate interior angles, making them useful in quick geometric reasoning.
Key Points to Remember
- Alternate exterior angles are on opposite sides of the transversal and outside the two lines.
- When two parallel lines are cut by a transversal, alternate exterior angles are equal.
- The converse is also true: if alternate exterior angles are equal, the lines are parallel.
- There are two pairs of alternate exterior angles when a transversal cuts two lines.
- Alternate exterior angles can be proved equal using corresponding angles + vertically opposite angles.
- Co-exterior angles (same-side exterior) are supplementary (sum = 180°) when lines are parallel.
- Do NOT confuse alternate exterior with alternate interior angles — exterior angles are OUTSIDE the two lines.
- This property is frequently used in proofs involving parallel lines, triangles, and quadrilaterals.
- The theorem does NOT apply if the lines are not parallel — the angles will generally be unequal.
- Standard angle labelling: number angles 1–4 at the first intersection and 5–8 at the second.
Practice Problems
- Lines AB ∥ CD are cut by transversal PQ. If ∠APQ = 132°, find the alternate exterior angle at CD.
- A transversal cuts two lines making alternate exterior angles of (5x − 8)° and (3x + 22)°. Find x and determine if the lines are parallel.
- Two parallel lines are cut by a transversal. One angle is 47°. Find all 8 angles formed.
- Prove that if two parallel lines are cut by a transversal, the co-exterior angles are supplementary.
- In a figure, line l ∥ line m. A transversal makes an angle of 68° with line l on the upper-left. Find the angle it makes with line m on the lower-right.
- Three parallel lines are cut by a transversal. If the exterior angle at the first line is 140°, find the corresponding alternate exterior angles at the other two lines.
- Lines PQ and RS are cut by transversal MN. ∠PMN = 108° and ∠RNM = 72°. Are PQ and RS parallel? Justify.
- Using the alternate exterior angles theorem, prove that the exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.
Frequently Asked Questions
Q1. What are alternate exterior angles?
Alternate exterior angles are pairs of angles formed when a transversal crosses two lines. They lie on opposite sides of the transversal and outside the two lines.
Q2. Are alternate exterior angles always equal?
They are equal only when the two lines are parallel. If the lines are not parallel, alternate exterior angles are generally not equal.
Q3. How do you identify alternate exterior angles?
Look for angles that are (1) on opposite sides of the transversal, (2) outside the two lines, and (3) at different intersections. There are exactly two such pairs.
Q4. What is the difference between alternate interior and alternate exterior angles?
Alternate interior angles are between the two lines (interior). Alternate exterior angles are outside the two lines (exterior). Both types are on opposite sides of the transversal, and both are equal when lines are parallel.
Q5. What are co-exterior angles?
Co-exterior (or same-side exterior or consecutive exterior) angles are on the same side of the transversal and outside the two lines. When the lines are parallel, co-exterior angles are supplementary (sum = 180°).
Q6. Can alternate exterior angles be used to prove lines parallel?
Yes. The converse of the theorem states: if a transversal cuts two lines such that a pair of alternate exterior angles is equal, then the lines are parallel.
Q7. How many pairs of alternate exterior angles are formed?
When a transversal intersects two lines, exactly two pairs of alternate exterior angles are formed.
Q8. Is the alternate exterior angles theorem in the NCERT Class 9 syllabus?
The NCERT Class 9 textbook covers the corresponding angles axiom and alternate interior angles theorem in detail. Alternate exterior angles follow as a direct consequence and are used in exercises and proofs.
Q9. How are alternate exterior angles proved equal?
Using two steps: (1) A corresponding angle pair is equal (by the corresponding angles axiom for parallel lines). (2) A vertically opposite angle pair is equal. Combining these gives alternate exterior angles equal.
Q10. Can a single transversal create more than two pairs of alternate exterior angles?
When a transversal cuts two lines, there are exactly two pairs. If the transversal cuts three parallel lines, there are two pairs at each pair of lines, giving more total pairs.
