Quick Revision

• When two lines intersect, they form four angles. Vertically Opposite Angles are equal and each Linear Pair sums to \(180^\circ\).
• Perpendicular lines meet at a right angle: \(90^\circ\).
• Parallel lines never meet on a plane. Notation: \( \ell \parallel m \). Perpendicular notation: \( \ell \perp m \).
• A transversal cuts two lines and creates 8 angles.
• If the lines are parallel:
  Corresponding angles are equal \((\text{F-shape})\).

  Alternate interior angles are equal \((\text{Z-shape})\).

  Co-interior (same-side interior) angles are supplementary: sum \(=180^\circ\) \((\text{C-shape})\).

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1. Define vertically opposite angles.
   When two lines intersect, each pair of opposite angles are equal; they are called vertically opposite angles.
2. If two angles form a linear pair, what is their sum?
   \(180^\circ\).
3. Two lines intersect to make one angle \(120^\circ\). What is the vertically opposite angle?
   \(120^\circ\).
4. Give the symbol used to denote parallel lines.
   \( \parallel \) (e.g., \( \ell \parallel m \)).
5. Give the symbol used to denote perpendicular lines.
   \( \perp \) (e.g., \( \ell \perp m \)).
6. What is a transversal?
   A line that cuts two (or more) lines at distinct points.
7. Name the property: If \( \ell \parallel m \), corresponding angles are ________.
   Equal.
8. If \( \ell \parallel m \), alternate interior angles are ________.
   Equal.
9. If \( \ell \parallel m \), the sum of interior angles on the same side of transversal is ________.
   \(180^\circ\).
10. Two lines intersect and one angle is \(65^\circ\). Find its adjacent angle.
    \(180^\circ - 65^\circ = 115^\circ\).
11. State whether the following is true/false: “Parallel lines meet at infinity.”
    False (in Euclidean plane geometry, they never meet).
12. The angle formed by a straight line is ________.
    \(180^\circ\).
13. If one angle of a perpendicular pair is \(90^\circ\), the other is ________.
    \(90^\circ\).
14. State any one method to check if two lines are parallel using a transversal.
    Show a pair of corresponding angles are equal.
15. Which figure indicates equal corresponding angles? (F/Z/C)
    F-shape.
16. Which figure indicates equal alternate interior angles? (F/Z/C)
    Z-shape.
17. Which figure indicates co-interior angles summing to \(180^\circ\)? (F/Z/C)
    C-shape.
18. Two intersecting lines form four equal angles. What are the lines called?
    Perpendicular lines.
19. If \( \angle a = 104^\circ \), find its vertically opposite angle.
    \(104^\circ\).
20. If a transversal cuts two lines such that a pair of corresponding angles are unequal, the lines are ________.
    Not parallel.

1. Two lines intersect. One angle is \(38^\circ\). Find all other three angles.
   Adjacent angles: \(180^\circ-38^\circ=142^\circ\). Vertically opposite: \(38^\circ\). The last one (adjacent to \(142^\circ\)) is \(38^\circ\).
2. At an intersection, \( \angle a = 120^\circ\). Find its linear-pair angle and the other two.
   Linear pair: \(60^\circ\). Vertically opposite to \(120^\circ\) is \(120^\circ\). Remaining is \(60^\circ\).
3. A transversal cuts parallel lines \( \ell \parallel m\). If a corresponding angle is \(72^\circ\), find: (i) the alternate interior angle, (ii) its co-interior partner.
   (i) Alternate interior \(=72^\circ\). (ii) Co-interior \(=180^\circ-72^\circ=108^\circ\).
4. In \( \ell \parallel m\), a co-interior angle is \(95^\circ\). Find its interior partner on same side.
   \(180^\circ-95^\circ=85^\circ\).
5. Prove \( \ell \parallel m\) if a pair of alternate interior angles are equal.
   When alternate interior angles are equal, by the Converse of the Alternate Interior Angles Theorem, the lines are parallel.
6. A transversal cuts \( \ell\) and \(m\). If one corresponding angle is \(x\) and its adjacent interior angle on the same line is \(x+30^\circ\), prove \( \ell \not\parallel m\).
   On the same intersection, linear pair must sum to \(180^\circ\): \(x+(x+30^\circ)=180^\circ \Rightarrow x=75^\circ\). But the corresponding angle at the other intersection must also be \(75^\circ\) for parallelism. If given unequal, then not parallel.
7. Find \(x\): two straight lines cross; one angle is \( (3x+6)^\circ\) and its adjacent angle is \( (7x-6)^\circ\).
   Linear pair \(=180^\circ\): \( (3x+6)+(7x-6)=180 \Rightarrow 10x=180 \Rightarrow x=18\).
8. If \( \angle A\) and \( \angle B\) are alternate interior angles made by a transversal with two lines and \( \angle A= \angle B\), what can you conclude?
   The two lines are parallel.
9. In \( \ell \parallel m\), a corresponding angle is \(126^\circ\). Find all eight angles (measure set).
   Equal set: \(126^\circ\) (2 pairs V.O.A., 2 pairs corresponding). Supplementary set: \(54^\circ\) (the other four). So, angles are four \(126^\circ\) and four \(54^\circ\).
10. A transversal cuts \( \ell \parallel m\). One angle at the top intersection is \(45^\circ\). Find the co-interior angle at the bottom intersection on the same side.
    \(180^\circ-45^\circ=135^\circ\).
11. Show that if one pair of corresponding angles are equal, then all four corresponding pairs are equal.
    From one equal pair, the adjacent ones follow by linear pairs, and vertical opposites carry equality across, giving equality to all four corresponding pairs.
12. Two lines are perpendicular. One angle is \( (2y+10)^\circ\). Find \(y\).
    Right angles: \(90^\circ\). So \(2y+10=90 \Rightarrow y=40\).
13. A line is perpendicular to a transversal. If the transversal also makes a \(30^\circ\) angle with a second line, are the two lines parallel?
    Yes. Perpendicular to the same line \(\Rightarrow\) both make \(90^\circ\) with it; equal corresponding angles \(\Rightarrow\) parallel.
14. Find the angle marked \(x\): a straight line is split by another into \(x^\circ\) and \((3x-12)^\circ\).
    Linear pair: \(x+3x-12=180 \Rightarrow 4x=192 \Rightarrow x=48^\circ\).
15. If \( \ell \parallel m\) and a transversal gives one angle \(87^\circ\), find the alternate exterior angle.
    \(87^\circ\) (equal).
16. In \( \ell \parallel m\), a corresponding angle is \(\frac{3}{5}\) of a straight angle. Find the acute set.
    \(\frac{3}{5}\times 180^\circ=108^\circ\). Supplement \(=72^\circ\). Acute angles are \(72^\circ\).
17. State the condition (using interior angles) that guarantees two lines are parallel.
    If same-side interior (co-interior) angles are supplementary (sum \(180^\circ\)), then lines are parallel.
18. A transversal cuts two lines. If one interior angle is \(x\) and the adjacent interior on same side is \(x\), what can you say about the lines?
    They cannot be parallel since co-interior angles must sum to \(180^\circ\), not be equal (unless \(x=90^\circ\) and still sum \(180^\circ\) would need the other \(=90^\circ\), but being adjacent interior on same side being equal does not alone prove parallel).
19. State a quick paper-folding test for perpendicularity.
    Fold to make one crease; fold again so the second crease is at a right angle to the first (edges aligned). The creases are perpendicular.
20. In \( \ell \parallel m\), if one angle is \( (5x-10)^\circ\) and its corresponding angle is \( (4x+20)^\circ\), find \(x\).
    Equal: \(5x-10=4x+20 \Rightarrow x=30\).

1. Lines \( \ell\) and \(m\) intersect. If one angle is \( (3x+9)^\circ\) and its adjacent angle is \( (5x-21)^\circ\), find all four angles.
   Linear pair: \(3x+9+5x-21=180 \Rightarrow 8x-12=180 \Rightarrow x=24\). Angles: \(3x+9=81^\circ\), adjacent \(=99^\circ\); vertically opposite are \(81^\circ\) and \(99^\circ\).
2. A transversal cuts \( \ell \parallel m\). One angle is \( (2y+6)^\circ\) and its alternate interior partner is \( (y+56)^\circ\). Find \(y\) and all the distinct angle measures.
   Equal: \(2y+6=y+56 \Rightarrow y=50\). So that angle is \(106^\circ\). Supplementary set \(=74^\circ\).
3. In a figure with \( \ell \parallel m\), a corresponding angle is twice its co-interior partner. Find both angles.
   Let corresponding \(=2a\), co-interior \(=a\), with \(2a+a=180 \Rightarrow a=60^\circ\). Hence \(120^\circ\) and \(60^\circ\).
4. Show that if one line is perpendicular to a transversal and the other line makes equal corresponding angles with the same transversal, then the two lines are parallel.
   Equal corresponding angles \(\Rightarrow\) parallel by converse; perpendicularity ensures one is \(90^\circ\) so the other corresponding is also \(90^\circ\), confirming parallelism.
5. In a “Z-shape” with \( \ell \parallel m\), the top interior angle is \( (x+25)^\circ\) and the bottom interior angle is \( (2x-5)^\circ\). Find \(x\) and both angles.
   Alternate interior equal: \(x+25=2x-5 \Rightarrow x=30\). Angles \(=55^\circ\).
6. A transversal cuts two lines so that a pair of co-interior angles are \( (3t+15)^\circ\) and \( (5t-5)^\circ\). If the lines are parallel, find \(t\) and the two angles.
   Sum \(180^\circ\): \(3t+15+5t-5=180 \Rightarrow 8t+10=180 \Rightarrow t=21.25\). Angles: \(3t+15=78.75^\circ\), \(5t-5=101.25^\circ\).
7. In intersecting lines, the four angles around are in the ratio \(2:3:2:3\). Find each angle.
   Adjacent sum \(180^\circ\): \(2k+3k=180^\circ \Rightarrow 5k=180^\circ \Rightarrow k=36^\circ\). So angles: \(72^\circ,108^\circ,72^\circ,108^\circ\).
8. Two lines are cut by a transversal. Show that if one exterior angle equals the interior opposite angle (not adjacent) then the lines are parallel.
   That pair are alternate exterior/interior; equality implies parallel by converse of the alternate angle theorem.
9. Angles around a point sum to \(360^\circ\). Three angles are \(x, \; (x+20)^\circ,\; (2x-10)^\circ\) and the fourth is a right angle. Find all.
   \(x+(x+20)+(2x-10)+90=360 \Rightarrow 4x+100=360 \Rightarrow x=65^\circ\). Angles: \(65^\circ,85^\circ,120^\circ,90^\circ\).
10. If two lines are parallel and a transversal makes one angle \(135^\circ\), find (i) its vertically opposite, (ii) adjacent, (iii) alternate interior angles.
    (i) \(135^\circ\). (ii) \(45^\circ\). (iii) \(135^\circ\).
11. Prove: If corresponding angles are equal, then alternate interior angles are equal.
    Corresponding equality \(\Rightarrow\) parallel. For parallel lines, alternate interior are equal. Hence proved.
12. A pair of lines cut by a transversal gives one angle \(x^\circ\). If the angle adjacent to its corresponding partner is \( (x+30)^\circ\), prove the lines are parallel and find \(x\).
    Adjacent to corresponding \(=180^\circ - x\). Given \(x+30 = 180-x \Rightarrow 2x=150 \Rightarrow x=75^\circ\). With a consistent set of equal corresponding angles, lines are parallel.
13. Two intersecting lines form angles \( (4y-6)^\circ\) and \( (2y+24)^\circ\) opposite each other. Find all angles.
    V.O.A equal: \(4y-6=2y+24 \Rightarrow 2y=30 \Rightarrow y=15\). Angles: \(54^\circ\) and \(126^\circ\) (the linear pair).
14. A transversal cuts \( \ell \parallel m\). One interior angle is 1.2 times its adjacent co-interior angle. Find both.
    Let angles be \(1.2a\) and \(a\), with sum \(180^\circ\): \(2.2a=180 \Rightarrow a=81.818\ldots^\circ\). So \(a=\frac{900}{11}^\circ\approx81.818^\circ\), other \(= \frac{1080}{11}^\circ\approx98.182^\circ\).
15. If a pair of lines are each perpendicular to a third line, prove they are parallel.
    Each makes a right angle with the third line; corresponding angles are equal \((90^\circ)\), hence the two lines are parallel.
16. In a figure, \( \angle A\) and \( \angle B\) are corresponding angles. If \( \angle A=(7z-8)^\circ\) and \( \angle B=(5z+32)^\circ\), find \(z\) and both angles.
    Equal: \(7z-8=5z+32 \Rightarrow 2z=40 \Rightarrow z=20\). Angles \(=132^\circ\).
17. A straight line is split by a crossing line into angles in the ratio \(3:2\). Find the two angles and all angles at the intersection.
    \(3k+2k=180 \Rightarrow k=36^\circ\). So \(108^\circ,72^\circ\). Other two are their vertical opposites.
18. Show that if a pair of alternate exterior angles are equal, then lines are parallel.
    Converse of the alternate (exterior) angle property gives parallelism.
19. In \( \ell \parallel m\), a corresponding angle is \(x\) and the angle adjacent to it at the same intersection is \( (x+20)^\circ\). Find both sets of measures.
    Linear pair: \(x+(x+20)=180 \Rightarrow x=80^\circ\). So equal set \(=80^\circ\), supplementary set \(=100^\circ\).
20. Two lines are cut by a transversal. If co-interior angles differ by \(30^\circ\), find both angles.
    Let them be \(a\) and \(a+30\) with sum \(180\): \(2a+30=180 \Rightarrow a=75^\circ\), other \(=105^\circ\).

5.1 Across the Line — Intersecting Lines

Q1. (Fig. 5.3) List all Linear Pairs and Vertically Opposite Angles when two lines \( \ell\) and \(m\) intersect to form angles \(a,b,c,d\) in order.

Linear Pairs: \((\angle a,\angle b),\; (\angle b,\angle c),\; (\angle c,\angle d),\; (\angle d,\angle a)\) each sums to \(180^\circ\).
Vertically Opposite Pairs: \((\angle a,\angle c)\) and \((\angle b,\angle d)\) (equal pairs).

Q2. If \( \angle a=120^\circ\) (as in Fig. 5.2-type intersection), find \( \angle b, \angle c, \angle d\).

\( \angle b=60^\circ\) (linear pair), \( \angle c=120^\circ\) (vertically opposite), \( \angle d=60^\circ\) (linear pair with \(c\)).

Q3. Define Linear Pair and Vertically Opposite Angles.

Linear Pair: adjacent angles whose arms form a straight line (sum \(180^\circ\)). Vertically Opposite: opposite angles at intersection; equal.

5.2 Perpendicular Lines

Q4. If two lines intersect and all four angles are equal, prove each is \(90^\circ\).

Let each be \(x\). Around a point: \(4x=360^\circ \Rightarrow x=90^\circ\). Hence the lines are perpendicular.

5.3 Between Lines (Identifying parallel/perpendicular)

Q5. Explain how to decide if two drawn segments are likely to meet when extended.

Check relative direction: if they preserve constant separation (equal corresponding angles with a transversal), they are parallel (won’t meet). Otherwise, if directions differ, they will meet.

5.4 Parallel & Perpendicular via Paper Folding

Q6. After horizontal folds on a square sheet, how many pairs of parallel lines are visible? What is their relation to vertical edges?

Each crease is parallel to the others (many parallel lines). All horizontal creases are perpendicular to the vertical edges.

Q7. Notation: How do we mark parallel and perpendicular in a figure?

Parallel sets: single, double, … arrow marks. Perpendicular: a small square at the right angle.

5.5 Transversals

Q8. When a transversal cuts two lines, how many angles form? How many distinct measures maximum?

Eight angles form. At most four distinct measures (due to vertical opposites and linear pairs).

5.6 Corresponding Angles

Q9. State the Corresponding Angles Theorem and its converse.

If a transversal cuts parallel lines, corresponding angles are equal. Converse: If a pair of corresponding angles are equal, the lines are parallel.

5.7 Drawing Parallel Lines

Q10. Using a set square, explain why two drawn lines are parallel when their corresponding angles with a given line are \(90^\circ\).

Each makes a right angle with the baseline; equal corresponding angles imply the lines are parallel.

Q11. Through point \(A\), draw a line parallel to \( \ell\) using a compass/straightedge. Outline steps.

(i) Draw a transversal through \(A\) cutting \( \ell\) at \(P\). (ii) Copy the angle formed at \(P\) onto \(A\) (corresponding angle construction) using compass arcs. (iii) The new line through \(A\) with equal corresponding angle is parallel to \( \ell\).

5.8 Alternate Angles & Angle Chasing

Q12. Prove alternate interior angles are equal when \( \ell \parallel m\).

From equal corresponding angles and vertical opposite equality, the interior alternates on the “Z-shape” are equal.

Q13. (Example 1) In Fig. 5.26, if \( \angle 6=135^\circ\), find all eight angles.

\( \angle 2=\angle 4=\angle 6=\angle 8=135^\circ\); the remaining four are \(45^\circ\).

Q14. (Example 2) In Fig. 5.27, \( \angle a=120^\circ\) and \( \angle f=70^\circ\). Are \( \ell\) and \(m\) parallel?

No. Corresponding angles should be equal; here they are not.

Q15. (Example 3) In Fig. 5.28, \( \angle 3=50^\circ\). Find \( \angle 6\).

\( \angle 2=130^\circ\) (linear with \(3\)). \( \angle 6=\angle 2=130^\circ\) (corresponding).

Q16. (Example 4) In Fig. 5.29, \(AB \parallel CD\) and \(AD \parallel BC\). Given \( \angle DAC=65^\circ\), \( \angle ADC=60^\circ\). Find \( \angle CAB,\; \angle ABC,\; \angle BCD\).

\( \angle DAB = 180^\circ - 60^\circ = 120^\circ\). Then \( \angle CAB = 120^\circ - 65^\circ = 55^\circ\). Also \( \angle BCD = 120^\circ\) (interior with \(ADC\)), and \( \angle ABC = 60^\circ\).

Angle-Find Practice (Guided Solutions for Figure Sets)

For the textbook’s diagram-specific angle grids (Figs. 5.30–5.35), apply exactly these three tools: (1) vertically opposite are equal, (2) linear pair \(=180^\circ\), (3) for parallels: corresponding/alternate equal, co-interior sum \(180^\circ\). Every marked angle follows by 2–3 such steps.

How to Show Work (Neat MathJax Formatting)

End Summary

• Intersecting lines: VOAs equal; linear pairs supplementary.
• Perpendicular lines: all four angles \(90^\circ\).
• Parallel lines & transversal: Corresponding \(=\), Alternate \(=\), Co-interior sum \(180^\circ\).
• Converse statements help prove lines are parallel.