Lines and angles

Look around you: the edges of a door, the lines on a badminton court, the frame of a bicycle. They are all made of lines that meet at angles. Builders use lines and angles to make structures strong, artists use them to make patterns, and land surveyors use them to find heights and distances. In this chapter you will measure and construct lines and angles, and learn the rules for the angles made by intersecting lines and by parallel lines.

8.1Lines and angles

Naming line segments and angles

An angle is named with the symbol ∠ and three capital letters. The middle letter is always the vertex (the corner where the two arms meet).

Congruent line segments and congruent angles

Estimating and measuring

To estimate a length, compare it with an object whose length you know. If an eraser is 4 cm long and a line is about 2½ erasers long, the line is about 4 + 4 + 2 = 10 cm. To measure it exactly, use a ruler, starting at the 0 mark.

To estimate an angle, compare it with a right angle (90°). A little less than a right angle is about 80°; about half a right angle is about 45°. To measure exactly, use a protractor.

Types of angles

Complementary, supplementary and conjugate angles

Example 1: angles at a point (textbook p. 175)

p and 54° are complementary. p and q are supplementary. The angle conjugate to r is 260°. Find p, q, r and s.

1. p + 54° = 90°, so p = 90° − 54° = 36°
2. p + q = 180°, so q = 180° − 36° = 144°
3. r + 260° = 360°, so r = 360° − 260° = 100°
4. Angles at a point: s + 100° + 144° + 36° + 54° = 360°
   s + 334° = 360°, so s = 26°

Geometrical constructions

A geometrical construction is an accurate drawing made with a pair of compasses and a straight edge (ruler). Use a sharp pencil and do not erase the construction arcs: they show your method.

Line segment AB = 8 cm

Draw a line and mark A. Open the compasses to 8 cm on a ruler. From A, mark an arc on the line: that point is B. This is more accurate than drawing along a ruler, because the compasses hold the distance.

Perpendicular bisector of PQ

1. Open the compasses to more than half of PQ.
2. Draw arcs above and below PQ from P, then from Q with the same radius.
3. Join the two crossing points. This line cuts PQ into two equal parts at 90°.

Perpendicular from a point M to a line PQ

1. From M, draw arcs cutting PQ at R and S.
2. From R and from S (same radius), draw arcs that cross on the other side of PQ.
3. Join M to the crossing point.

For a point N on the line, do the same: mark R and S on either side of N, then widen the compasses and draw crossing arcs from R and S.

Line through R parallel to PQ

1. With radius PR, draw an arc from Q.
2. With radius PQ, draw an arc from R. The two arcs cross at S.
3. Join R and S. PQSR is a parallelogram, so RS is parallel to PQ.

Angle of 60°

1. Draw a long arc from Q cutting QR at S.
2. With the same radius, draw an arc from S cutting the first arc at P.
3. Join QP. QS = SP = QP, so triangle PQS is equilateral and ∠PQR = 60°.

Angle bisector

1. Draw an arc from Q cutting both arms at M and N.
2. With the same radius, draw arcs from M and from N that cross at S.
3. Join QS. Bisecting 60° gives ∠PQS = ∠SQR = 30°.

Start with 60° and keep bisecting or adding:

8.2Angles related to intersecting lines

Example 10: finding x and y (textbook p. 186)

PSQ, RSTU and PTV are straight lines. Find x and y.

1. x and 135° are adjacent angles on the straight line RSU: x + 135° = 180°, so x = 45°.
2. y and 62° are vertically opposite angles: y = 62°.

Example 11: solving a problem (textbook p. 187)

POR and TOQ are straight lines. Find x and y.

1. ∠POT and ∠QOR are vertically opposite, so ∠POT = 72°.
2. x + 60° = 72°, so x = 12°.
3. ∠TOR and ∠QOR are adjacent: ∠TOR = 180° − 72° = 108°.
4. y + y = 108°, 2y = 108°, so y = 54°.

8.3Angles related to parallel lines and transversals

Are the lines parallel?

Turn the rules round: if corresponding angles are equal, or alternate angles are equal, or interior angles add up to 180°, then the lines are parallel.

Example 12 (textbook p. 190)

1. The corresponding angles are both 48°, so PQ // RS.
2. 42° and 128° are interior angles: 42° + 128° = 170° ≠ 180°. So PQ and RS are not parallel.

Finding unknown angles

Worked example: finding a, b and c (like Example 13, textbook p. 191)

PQ // RS. Find a, b and c. Always give the reason.

1. a and 118° are interior angles: a = 180° − 118° = 62°.
2. b and 118° are alternate angles: b = 118°.
3. c and 118° are corresponding angles: c = 118°.

Worked example: the zigzag (like Self Practice 8.3d, textbook p. 191)

AB // CD. Find x.

1. Draw a helper line through E parallel to AB (and CD).
2. The top part of x and 30° are alternate angles, so it is 30°.
3. The bottom part of x and 45° are alternate angles, so it is 45°.
4. x = 30° + 45° = 75°.

Angles of elevation and depression

Summary