May 2013
Grade 9
Geometry of Straight Lines
Goals:
□ Write clear descriptions of the relationships between angles formed by:
o Perpendicular Lines
o Intersecting Lines
o Parallel Lines cut by a Transversal
□ Solve equations and missing angle values from intersecting lines.
(Note: Due to time constraints, we will spend more time solving equations with
straight lines in terms 3 and 4.)
□ Give justification when solving geometric problems.
Terminology










Straight Angle
Intersecting Lines
Parallel Lines
Perpendicular Lines
Transversal
Vertically Opposite Angles
Adjacent Angles
Alternate Angles
Corresponding Angles
Co-Interior Angles
Terminology
1 Line. Any angle on a single line is called a straight
angle. Straight angles have a measure of 180°.
E.g. Ĉ = 180°.
2 Lines. Any two lines can relate in the following ways:
1) Parallel Lines – Lines which do not intersect are
called parallel lines. We use the symbol ∥ to denote
parallel lines.
E.g. 𝑝 ∥ 𝑞.
2) Intersecting Lines – Lines which intersect at a
point of intersection are called intersecting lines.
E.g. 𝑞 intersects 𝑝 at point A.
3) Perpendicular Lines – Lines which
intersect at a 90° angle are called
perpendicular lines. We use the symbol ⊥
to denote perpendicular lines.
E.g. 𝑞 ⊥ 𝑝.
When 2 Lines intersect, the following angle relationships are formed:
Vertically Opposite Angles – Angles which are on opposite sides of 2 intersecting
lines are called vertically opposite angles. Vertically opposite angles are congruent.
E.g. ∠1 and ∠3 are vertically opposite.
∴ ∠1 ≡ ∠3
Adjacent Angles – Angles which share a common edge are
called adjacent angles. Adjacent angles formed by
intersecting lines are supplementary, which means they sum
to 180°.
E.g. ∠2 and ∠3 are adjacent angles.
∴ ∠2 and ∠3 are supplementary, i.e. ∠2 + ∠3 = 180°
3+ Lines. A line which intersects two (or more) other lines is called a transversal. E.g. line 𝑠
transverses lines 𝑝 and 𝑞 below. Transversals create a significant number of interesting
angle relationships, which are identified below.
1) Alternate Angles – Interior angles on alternating sides of the transversal are called
alternate angles. In the diagram below, the following are alternate angles:
∠2 and ∠8 ; ∠4 and ∠5
2) Corresponding Angles – The following are corresponding angles:
∠1 and ∠5 ; ∠2 and ∠6 ; ∠3 and ∠7 ; ∠4 and ∠8
3) Co-Interior Angles – Interior angles on the same side of the transversal are called
co-interior angles. The following are co-interior angles:
∠4 and ∠8; ∠2 and ∠5
If lines 𝒑 and 𝒒 are parallel (i.e. 𝑝 ∥ 𝑞), then the following is also true:
1) Alternate angles are congruent. E.g. ∠2 ≡ ∠8
2) Corresponding angles are congruent. E.g. ∠3 ≡ ∠8
3) Co-Interior angles are supplementary. ∠4 and ∠8 are supplementary
Exercise 1
1. Use the diagram to the right:
̂1?
1.1. Which angle is vertically opposite to E
̂ 4?
1.2. Which angle is vertically opposite to D
̂ 2?
1.3. Which two angles are adjacent to D
̂3 ?
1.4. Which two angles are adjacent to E
̂4
1.5. Which angle is an alternate angle to E
̂1
1.6. Which angle is an alternate angle to D
̂4 ?
1.7. Which angle corresponds to E
̂ 3?
1.8. Which angle corresponds to D
̂3 ?
1.9. Which angle is co-interior with E
̂1?
1.10. Which angle is co-interior with D
1.11. Which line is the transversal?
̂3 ≡ E
̂1? Why or why not?
1.12. Is E
̂3 ≡ E
̂1? Why or why not?
1.13. Is E
2. Fill in the blanks. Each question has exactly one correct answer. Justifications are written
in brackets
̂ 3 ≡ _____ (vertical angles)
2.1. H
̂3 + H
̂ 2 = ______ (adjacent angles)
2.2. H
2.3. Ĵ3 ≡ _____ (vertical angles)
̂ 1 + _____ = 180° (co-interior angles)
2.4. H
̂ 1 ≡ _____ (alternate angles)
2.5. H
2.6. _____ ≡ Ĵ4 (vertical angles)
2.7. Ĵ3 + _____ = 180° (co-interior angles)
2.8. _____ ≡ Ĵ2 (alternate angles)
̂1 ≡ H
̂ 3 (_______________)
2.9. H
̂1 + H
̂ 4 = 180° (_______________)
2.10. H
̂ 1 + Ĵ2 = 180° (_______________)
2.11. H
2.12. Ĵ1 + Ĵ2 = 180° (_______________)
Grade 9: Geometry of Straight Lines
̂1 = 70° and 𝑝 ∥ 𝑞, then determine the
3. If V
following angle measures. Give a justification
for every answer (i.e. state the angle
relationship you used.)
̂4 = _____ (_______________)
3.1. V
̂3 = _____ (_______________)
3.2. V
̂2 = _____ (_______________)
3.3. V
̂ 4 = _____ (_______________)
3.4. W
̂ 3 = _____ (_______________)
3.5. W
̂ 2 = _____ (_______________)
3.6. W
̂ 1 = _____ (_______________)
3.7. W
4. Give all possible examples of the following in
the diagram to the right.
4.1. Alternate Angles
4.2. Co-Interior Angles
4.3. Corresponding Angles
̂ 1 = 30° and B
̂1 = 120°, find the missing angle measures:
5. If A
̂2
5.1. A
̂3
5.2. A
̂4
5.3. A
̂2
5.4. B
̂3
5.5. B
̂4
5.6. B
5.7. Ĉ1
5.8. Ĉ2
5.9. Ĉ3
5.10. Ĉ4
5.11. Ĉ5
5.12. Ĉ6
5
Grade 9: Geometry of Straight Lines
̂ 1 = 30° and B
̂1 = 120°, find the missing angle measures:
6. If A
̂2
6.1. A
̂3
6.2. A
̂4
6.3. A
̂2
6.4. B
̂3
6.5. B
̂4
6.6. B
6.7. Ĉ1
6.8. Ĉ2
6.9. Ĉ3
6.10. Ĉ4
6.11. Ĉ5
6.12. Ĉ6
̂ 2 = 98° and M
̂ 4 = 110°. Find
7. Assume N
the missing angle measures.
7.1. ̂
P1
̂4
7.2. M
̂2
7.3. M
̂2
7.4. P
̂1
7.5. N
̂3
7.6. N
̂3
7.7. O
̂1
7.8. O
̂4
7.9. O
7.10. What shape is quad MPNO?
̂ 1 = 50°.
8. Assume W
̂ 3 and why?
8.1. What is the measure of W
8.2. What shape is quad WXYZ?
8.3. Hence what is the measure of Ẑ1 ?
8.4. What is the measure of Ẑ3 and why?
8.5. Using the first four parts of
this question, write a proof
which clearly and carefully
̂ 1 ≡ Ẑ3.
explains why W
6
Grade 9: Geometry of Straight Lines
9. Algebra with straight lines:
̂ 1 = 130° and H
̂ 4 = 6𝑥 − 10, solve for 𝑥.
9.1. If H
̂ 2 = 125° and H
̂ 4 = 8𝑡 + 15, solve for 𝑡.
9.2. If H
9.3. If Ĵ4 = 2𝑦 + 11 and Ĵ2 = 𝑦 + 7, solve for 𝑦.
̂ 1 = 130 + 2𝑎, solve for 𝑎.
9.4. If Ĵ2 = −2𝑎 + 50 and H
̂ 1 = 4𝑥 and Ĵ2 = 𝑥, what is the measure of H
̂ 1?
9.5. If H
̂ 2 = 45 − 3𝑥, what is the measure of H
̂ 1?
9.6. If If Ĵ4 = 8𝑥 − 10 and H
7