Concept used. Two distinct straight lines can have at most one common point. When two lines intersect, adjacent angles form linear pairs and opposite angles are vertically opposite angles.

1. Two different straight lines cannot meet at two points. If they had two common points, the same straight line would pass through those two points.
2. So two straight lines either do not meet or intersect at exactly one point.
3. If the four angles around the intersection are a,b,c,d, then a+b=180^∘ and b+c=180^∘ because each pair makes a straight angle.
4. From these relations, a=c and b=d. These are the vertically opposite angle pairs.

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Concept used. A linear pair consists of adjacent angles on a straight line. Vertically opposite angles lie opposite each other when two lines intersect.

1. ∠ a and ∠ b are adjacent and form a straight angle, so they are a linear pair.
2. ∠ b and ∠ c form another straight angle.
3. ∠ c and ∠ d form another straight angle.
4. ∠ d and ∠ a also form a straight angle.
5. Opposite angles are ∠ a and ∠ c, and ∠ b and ∠ d.

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Concept used. Parallel lines lie on the same plane and do not meet however far we extend them.

1. Lines a, i and h are drawn in the same vertical direction, so they appear parallel.
2. Lines c and g are both horizontal, so they appear parallel.
3. Lines d and f have the same slant, so they appear parallel.
4. Lines e and b also have the same slant, so they appear parallel.

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Concept used. Opposite edges of a square sheet are parallel. Adjacent edges are perpendicular because they meet at right angles.

1. The opposite edges of the square sheet are parallel to each other.
2. The adjacent edges are perpendicular to each other because they meet at right angles.
3. After one horizontal fold, the two horizontal edges and the crease are parallel. So we see 3 parallel horizontal lines.
4. After one more horizontal fold, the visible horizontal lines become 5.
5. After another horizontal fold, the visible horizontal lines become 9.
6. A vertical fold is perpendicular to the previous horizontal fold lines.
7. A fold parallel to a diagonal can be made by aligning the fold with the same diagonal direction.

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Concept used. If two lines are made by the same folding direction, or if both are perpendicular to the same line, they are parallel.

1. Line segment a and line segment p lie along the same fold direction, so they are parallel.
2. Line segment b and line segment q are both formed in matching positions with respect to the central crease.
3. Since their directions match, b ∥ q.
4. The triangular folds create c and r along matching diagonal directions.
5. Hence c ∥ r as well.

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Concept used. Perpendicular lines meet at 90^∘. Parallel lines keep the same direction and distance when extended.

1. In Fig. 5.10, draw each required perpendicular by making a right angle with the given line segment on the dot paper.
2. In Fig. 5.11, mark perpendicular lines where a vertical-looking line meets a horizontal-looking line at a right angle.
3. Mark parallel lines in Fig. 5.11 by using the same arrow notation for lines with the same direction.
4. In Fig. 5.12, draw each parallel through dot endpoints by keeping the same tilt as the given line segment.
5. The more difficult ones are usually e, f, g and h, because they are not simple horizontal or vertical segments.
6. In Fig. 5.13, line c is parallel to line a, not line b.

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Concept used. Vertically opposite angles are equal whenever two lines intersect.

1. At the upper intersection, ∠ 1=∠ 3 and ∠ 2=∠ 4.
2. At the lower intersection, ∠ 5=∠ 7 and ∠ 6=∠ 8.
3. Since at least these pairs are equal, all eight angles cannot have different measurements.
4. Among angles 6,5,4,3,2, the pair ∠ 2 and ∠ 4 is equal.
5. Therefore those five angles also cannot all be different.

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Concept used. If two lines are perpendicular to the same line, they are parallel to each other.

1. Place one side of a set square along line l.
2. Hold a ruler against another side of the set square so the ruler stays fixed.
3. Slide the set square along the ruler until its drawing edge passes through point A.
4. Draw the line through A. This line is parallel to l because the set square keeps the same angle while sliding.
5. In paper folding, first fold a crease t perpendicular to l through A.
6. Then fold another crease m perpendicular to t through A.
7. Since l and m are both perpendicular to t, they are parallel.

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Concept used. Use vertically opposite angles, linear pairs, corresponding angles and same-side interior angle sums.

1. In the first small figure, a=48^∘ by vertically opposite angles.
2. In the second, b=52^∘ by the corresponding or equal angle relation shown.
3. In the third, c=81^∘ because the given 99^∘ and 81^∘ form a straight angle pair.
4. In the fourth, d=99^∘ by vertically opposite or corresponding angle equality.
5. In the fifth, e=69^∘ by the equal angle relation in the parallel-line setup.
6. In the sixth, f=48^∘ because it is supplementary to 132^∘.
7. In the seventh, g=122^∘ by corresponding or vertically opposite equality.
8. In the eighth, h=75^∘ by the marked parallel-line angle relation.
9. In the ninth, i=54^∘.
10. In the tenth, j=97^∘.

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Concept used. Around parallel lines and transversals, use supplementary linear pairs and equal corresponding or alternate angles.

1. In the first diagram, the marked angle related to 42^∘ gives a=180^∘-42^∘=138^∘.
2. In the second diagram, the corresponding setup gives a=180^∘-62^∘=118^∘.
3. In the third diagram, combining the shown linear-pair relation gives a=105^∘.
4. In the fourth diagram, the relevant difference in the angle chase gives a=23^∘.

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Concept used. Interior angles on the same side of a transversal add to 180^∘, and alternate or corresponding angles can be equal when lines are parallel.

1. In the first figure, angle x works with the given 65^∘ angle and the parallel-line relation, giving x=25^∘.
2. The angle y is supplementary to 25^∘.
3. Therefore, y=180^∘-25^∘=155^∘.
4. In the second figure, the same angle-chasing relation gives x=25^∘.

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Concept used. Use corresponding and alternate angles on parallel lines, then use angles on a straight line.

1. The angle made by the right slanting line with the horizontal is 45^∘.
2. The corresponding angle at E gives ∠ GEH=45^∘.
3. The other slanting line gives ∠ FED=78^∘ by the corresponding angle relation.
4. Angles ∠ GEH, ∠ HEF and ∠ FED lie along the straight line through G,E,D.
5. Hence 45^∘+∠ HEF+78^∘=180^∘.
6. So ∠ HEF=180^∘-123^∘=57^∘.

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Concept used. If two lines are parallel to the same line, they are parallel to each other. A linear pair adds to 180^∘.

1. Since AB∥ CD and CD∥ EF, we also have AB∥ EF.
2. The given angle with EF is 55^∘.
3. The adjacent angle on the straight line is 180^∘-55^∘=125^∘.
4. The corresponding angle y equals this angle, so y=125^∘.
5. The angle x corresponds to the same obtuse angle because AB, CD and EF are parallel.
6. Hence x=125^∘ too.

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Concept used. Draw helper lines parallel to the given parallel lines. Then transfer angles using corresponding and alternate angle rules.

1. Draw a line through N parallel to LM and a line through O parallel to PQ, as suggested in the hint.
2. The given 40^∘ and 52^∘ angles can then be transferred to the broken path using parallel-line angle rules.
3. The angle at N marked 96^∘ fixes the turn of the middle segment.
4. After transferring the known angles and subtracting from the relevant straight-angle sums, the angle at O between ON and OP is 108^∘.

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