Theorem 5 Two lines are parallel if and only if for any transversal, corresponding angles are equal.
C A B D F H A E G
Theorem 6 Each exterior angle of a triangle is equal to the sum of the interior opposite angles.
B
Interior opposite angles
|∠X| = |∠A| + |∠B| X
Converse of Theorem 5 When the two lines being crossed by a transversal are parallel, the corresponding angles are equal.
Theorem 9 In a parallelogram, opposite sides are
equal and opposite angles are equal. A
B Theorem 10 The diagonals of a parallelogram
bisect one another. A
E D C D C
Converse 1 of Theorem 9 If the opposite angles of a quadrilateral are equal, then it is a parallelogram. Converse 2 of Theorem 9 If the opposite sides of a quadrilateral are equal, then it is a parallelogram.
Theorem 14 (Pythagoras) In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Theorem 15 (Converse of Pythagoras) If the square of one side of a triangle is the sum of the squares of the other two, then the angle opposite the first side is a right angle.
Corollary 3 Each angle in a semicircle is a right angle.
Corollary 4 If the angle standing on a chord at some point of a circle is a right angle, then the chord is a diameter.
A Hypotenuse = c b a c2 = a2 E 90° O 180° You should revise these theorems before you attempt the practice questions below. Practice questions 30.3
1. Calculate the missing angles in each of the following. Justify your answers in each case. A
(i)
C D 124°
(iv) A 30° D 474 Linking Thinking 1 B (v)
112° B C D
(ii) A 100° C D (vi) A C B 144° (iii) A C 149° D C + b2 B