CBSE Class 9 Mathematics Plane Geometry I Notes Set 11

Lines and Angles

A ray starts at a point and runs on forever in one direction; two rays with a common starting point (the vertex) form an angle. Angles are measured in degrees, with 1° = 60′ and 1′ = 60″.
Types by size: acute (less than 90°), right (exactly 90°), obtuse (between 90° and 180°), straight (180°) and reflex (between 180° and 360°).
Complementary angles add to 90° and supplementary angles add to 180°. Two adjacent angles whose outer arms form a straight line make a linear pair, so they are supplementary.
Vertically opposite angles, formed when two lines cross, are always equal.

Parallel Lines and a Transversal

When a transversal cuts two parallel lines: corresponding angles are equal, alternate interior angles are equal, and co-interior (consecutive interior) angles add to 180°.
The converse also holds: if any one of these relations is true, the two lines are parallel.
Useful trick: when a figure has no obvious transversal, draw an extra line through a vertex parallel to the given parallel lines.

Polygons and Clock Angles

The angle sum of an n-sided polygon is (n − 2) × 180°, and the exterior angles of any convex polygon add to 360°. A regular polygon has each exterior angle 360°/n. A polygon with n sides has n(n − 3)/2 diagonals.
Clock problems: the minute hand moves 6° per minute and the hour hand 0.5° per minute, so the minute hand gains 5.5° per minute. The angle at h hours m minutes is |30h − 5.5m|; if it exceeds 180°, subtract it from 360°.
The hands coincide 11 times in 12 hours and are opposite each other 11 times in 12 hours.

Illustration 1. Find the angle whose complement is one-third of its supplement.
Answer:
Let the angle be x. Then 90 − x = (180 − x)/3, so 270 − 3x = 180 − x.
2x = 90, so x = 45°.

Illustration 2. The interior angles of a convex polygon add to 1620°. How many sides has it?
Answer:
(n − 2) × 180° = 1620°, so n − 2 = 9.
n = 11.

Illustration 3. At what time between 4 and 5 o'clock are the hands of a clock together?
Answer:
At 4 o'clock the minute hand is 120° behind the hour hand. It gains 5.5° per minute.
t = 120 ÷ 5.5 = 240/11 minutes.
The hands coincide at 21 9/11 minutes past 4.

Triangles: Basic Facts

The three angles of a triangle add to 180°. An exterior angle equals the sum of the two interior angles that are not next to it.
In any triangle, a larger angle faces a longer side. The sum of any two sides exceeds the third side, and the difference of two sides is less than the third.
An angle bisector from one vertex divides the opposite side in the ratio of the other two sides (angle bisector theorem).
A median joins a vertex to the mid-point of the opposite side. The three medians meet at the centroid, which divides each median in the ratio 2 : 1 from the vertex. A median splits a triangle into two triangles of equal area.
Apollonius' theorem: AB² + AC² = 2(AD² + BD²) where AD is the median to BC. The sum of the squares of the three medians is three-fourths of the sum of the squares of the sides.

Congruence of Triangles

Two triangles are congruent when they match exactly in shape and size. They are congruent if any of the following holds:
SAS: two sides and the angle between them are equal.
ASA or AAS: two angles and a side are equal, with the side in either position.
SSS: all three sides are equal.
RHS: for right triangles, the hypotenuse and one other side are equal.
SSA (two sides and a non-included angle) is not a valid test in general. Once triangles are shown congruent, all their remaining parts are equal (CPCT).

Similar Triangles

Two triangles are similar if their corresponding angles are equal and their corresponding sides are in the same ratio. Tests: AA (two equal angles), SAS (one equal angle and the sides around it in proportion) and SSS (all three sides in proportion).
Basic proportionality (Thales) theorem: a line parallel to one side of a triangle that cuts the other two sides divides them in the same ratio. Conversely, a line dividing two sides in the same ratio is parallel to the third side.
The areas of similar triangles are in the ratio of the squares of corresponding sides, while perimeters, medians, altitudes and angle bisectors are in the same ratio as the sides.
The altitude on the hypotenuse of a right triangle splits it into two triangles similar to each other and to the whole triangle. This gives h² = pq, a² = pc and b² = qc, where p and q are the parts of the hypotenuse c.

Pythagoras Theorem

In a right triangle the square of the hypotenuse equals the sum of the squares of the other two sides. The converse is true: if c² = a² + b², the angle opposite c is a right angle.
If c² < a² + b² for the longest side c, the triangle is acute. If c² > a² + b², it is obtuse.
Common triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 9-40-41. In a right triangle the median to the hypotenuse equals half the hypotenuse.

Illustration 4. The sides of a triangle are 13 cm, 14 cm and 15 cm. Find the altitude on the side 14 cm.
Answer:
Semi-perimeter s = 21. Area = √(21 × 8 × 7 × 6) = 84 cm².
Altitude = 2 × 84 ÷ 14 = 12 cm.

Illustration 5. In a triangle ABC, DE ∥ BC with D on AB and E on AC. If AD = 4 cm, DB = 6 cm and AE = 6 cm, find EC.
Answer:
By the basic proportionality theorem, AD/DB = AE/EC.
4/6 = 6/EC, so EC = 9 cm. EC = 9 cm.

Illustration 6. The area of two similar triangles are 81 cm² and 49 cm². If a side of the smaller triangle is 14 cm, find the corresponding side of the larger one.
Answer:
Ratio of sides = √(81/49) = 9/7.
Side = 14 × 9/7 = 18 cm.

Illustration 7. In a triangle ABC, AD bisects ∠A and meets BC at D. If AB = 10, AC = 6 and BC = 12, find BD.
Answer:
BD/DC = AB/AC = 10/6 = 5/3 and BD + DC = 12.
BD = 12 × 5/8 = 7.5.

Illustration 8. The medians of a right triangle drawn from the ends of the hypotenuse have lengths 5 and √40. Find the hypotenuse.
Answer:
Let the legs be a and b. The two medians satisfy (a² + b²/4) + (b² + a²/4) = 25 + 40 = 65.
(5/4)(a² + b²) = 65, so a² + b² = 52.
Hypotenuse = √52 = 2√13.

Quadrilaterals

The angles of a quadrilateral add to 360°. In a parallelogram the opposite sides are equal and parallel, the opposite angles are equal, the adjacent angles add to 180° and the diagonals bisect each other. A diagonal splits it into two congruent triangles.
A quadrilateral is a parallelogram if (any one of): both pairs of opposite sides are equal; both pairs of opposite angles are equal; the diagonals bisect each other; or one pair of opposite sides is equal and parallel.
Rectangle: a parallelogram with a right angle, so its diagonals are equal. Rhombus: a parallelogram with equal sides, so its diagonals are perpendicular bisectors of each other. Square: both a rectangle and a rhombus; its diagonals are equal and perpendicular.
Trapezium: one pair of parallel sides. If the legs are equal (isosceles trapezium) the base angles and the diagonals are equal. The segment joining the mid-points of the legs is parallel to the bases and equals half their sum. The segment joining the mid-points of the diagonals equals half the difference of the bases.
Parallelogram law: the sum of the squares of the diagonals equals twice the sum of the squares of two adjacent sides.

Mid-point Theorem and Areas

The line joining the mid-points of two sides of a triangle is parallel to the third side and half as long. Conversely, a line through the mid-point of one side parallel to a second side bisects the third side.
The mid-points of the sides of any quadrilateral form a parallelogram (the Varignon parallelogram). The medial triangle of a triangle has one-fourth of its area.
Triangles on the same base and between the same parallels have equal areas. A parallelogram and a triangle on the same base and between the same parallels: the triangle has half the area. Triangles with the same height have areas in the ratio of their bases.

Illustration 9. The diagonals of a rhombus are 16 cm and 30 cm. Find its side and area.
Answer:
The diagonals bisect each other at right angles, giving right triangles with legs 8 and 15.
Side = √(64 + 225) = 17 cm. Area = ½ × 16 × 30 = 240 cm². Side 17 cm, area 240 cm².

Illustration 10. In a parallelogram ABCD the adjacent angles ∠A and ∠B are in the ratio 2 : 3. Find all four angles.
Answer:
Adjacent angles add to 180°, so ∠A = 72° and ∠B = 108°.
∠A = ∠C = 72° and ∠B = ∠D = 108°.

Illustration 11. The parallel sides of a trapezium are 12 cm and 20 cm. Find the length of the segment joining the mid-points of its diagonals.
Answer:
Segment = ½ (20 − 12) = 4 cm.