Pariksha SaathiLesson 5 of 8 · Mathematics, Classes VI-VIII (Paper II)
Geometry: Lines, Angles, Parallel Lines, Triangles and Congruence
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Basic geometrical ideas
A line segment has two end points and a fixed length, a ray has one end point, and a line extends endlessly both ways; exactly one line passes through two given points. Angles are acute (less than 90°), right (90°), obtuse (between 90° and 180°), straight (180°), reflex (between 180° and 360°) or complete (360°). The hour hand turns 30° an hour, so the hands make 60° at 2 o'clock and 120° at 4 o'clock.
Pairs of angles
Two angles are complementary if they add up to 90° and supplementary if they add up to 180°: the complement of 63° is 27° and its supplement 117°. Adjacent angles share a vertex and an arm without overlapping; a linear pair is a pair of adjacent angles whose other arms form a straight line, so it adds up to 180°. When two lines cross, the vertically opposite angles are equal. Two acute angles cannot be supplementary, and the angle equal to its complement is 45°. Supplementary angles in the ratio 4 : 5 are 80° and 100°, and an angle 30° more than its complement satisfies x = (90° − x) + 30°, so x = 60°.
Parallel lines and a transversal
A transversal cuts two or more lines at distinct points, and with two lines it makes eight angles. When the lines are parallel, corresponding angles are equal, alternate interior angles are equal, alternate exterior angles are equal, and interior angles on the same side of the transversal add up to 180°. The converses test for parallel lines: if a pair of corresponding or alternate angles is equal, or a pair of interior angles on the same side is supplementary, the lines are parallel. So if one of the eight angles is 70°, four are 70° and four are 110°; and if the interior angles on one side are 3x and 2x, then 5x = 180°, x = 36°, and the angles are 108° and 72°.
Triangles
By sides a triangle is scalene, isosceles or equilateral; by angles, acute, right or obtuse. A median joins a vertex to the mid-point of the opposite side and always lies inside; an altitude is the perpendicular from a vertex to the opposite side (or its extension) and can fall outside an obtuse triangle; every triangle has 3 of each. The angles add up to 180°, and an exterior angle equals the sum of the two interior opposite angles, so an exterior angle of 115° with one interior opposite angle of 50° leaves 65° for the other. The base angles of an isosceles triangle are equal, so a vertex angle of 40° leaves 70° for each. Any two sides together are longer than the third, so 2 cm, 3 cm and 6 cm cannot form a triangle, and with sides of 12 cm and 8 cm the third lies between 4 cm and 20 cm. The largest angle is opposite the longest side.
Pythagoras property
In a right-angled triangle the square on the hypotenuse equals the sum of the squares on the other two sides, h² = a² + b²; the Baudhayana Sulba Sutra states the same result for the diagonal of a rectangle. The converse decides whether a triangle is right-angled: 6, 8 and 10 satisfy 36 + 64 = 100, but 6, 8 and 11 do not. Common triplets are 3, 4, 5; 5, 12, 13; 8, 15, 17 and 7, 24, 25. A 15 m ladder reaching a window 12 m high stands 9 m from the wall, since 15² − 12² = 81; and a rhombus with diagonals of 16 cm and 30 cm, which bisect each other at right angles, has side √(8² + 15²) = 17 cm and perimeter 68 cm.
Congruence
Congruent figures have the same shape and size. Writing ΔABC ≅ ΔPQR also fixes the correspondence A ↔ P, B ↔ Q, C ↔ R, from which AB = PQ, ∠B = ∠Q and so on. The class VII criteria are SSS (three sides), SAS (two sides and the included angle), ASA (two angles and the included side) and RHS (right angle, hypotenuse and one side); since the third angle follows from the other two, two angles and any corresponding side also suffice. AAA is not a criterion, because triangles with equal angles can differ in size, and two sides with an angle that is not between them do not in general fix a triangle. If AB = DE = 5 cm, ∠B = ∠E = 60° and BC = EF = 7 cm, then ΔABC ≅ ΔDEF by SAS.
In the classroom
Pupils who read the wrong scale of a protractor report 140° for a 40° angle, believe an altitude must lie inside the triangle, or mix up complementary and supplementary. Estimating before measuring (is it more or less than a right angle?), folding paper to make a perpendicular bisector, and tearing off the three corners of a paper triangle to place them together on a straight line let pupils discover each property before it is stated.
Key facts
- The complement of 63° is 27° and its supplement 117°; supplementary angles in the ratio 4 : 5 are 80° and 100°.
- Clock: the hour hand moves 30° an hour, so the hands make 60° at 2 o'clock and 90° at 3 o'clock.
- Parallel lines cut by a transversal: corresponding, alternate interior and alternate exterior angles are equal; interior angles on the same side add up to 180° (3x + 2x = 180° gives 108° and 72°).
- Exterior angle property: an exterior angle of 115° with one interior opposite angle of 50° gives the other interior opposite angle as 65°.
- Triangle inequality: 2 cm, 3 cm and 6 cm cannot make a triangle (2 + 3 < 6); with sides 12 cm and 8 cm the third side is more than 4 cm and less than 20 cm.
- Pythagorean triplets: 3, 4, 5; 5, 12, 13; 8, 15, 17; 7, 24, 25 (and multiples such as 6, 8, 10).
- Ladder: a 15 m ladder reaching a window 12 m high has its foot √(225 − 144) = √81 = 9 m from the wall.
- Rhombus with diagonals 16 cm and 30 cm: side = √(64 + 225) = √289 = 17 cm, perimeter 68 cm.
- Congruence criteria: SSS, SAS, ASA and RHS; AAA, and two sides with a non-included angle, are not criteria.
- Every triangle has 3 medians and 3 altitudes; the medians lie inside, but two altitudes of an obtuse triangle fall outside it.
Quick revision
Complementary 90°, supplementary 180°; linear pair 180°; vertically opposite angles equal; with parallel lines, corresponding and alternate angles equal, same-side interior angles add to 180°; triangle angle sum 180°; exterior angle = sum of interior opposite angles; any two sides > third; h² = a² + b²; triplets 3-4-5, 5-12-13, 8-15-17, 7-24-25; congruence by SSS, SAS, ASA, RHS, never AAA.