Answer: Number 1 (a)
a = 22.5 Hence 2a = 45 and 3a = 67.5
Number 1 (b)
b = 244 and c = 26
Number 2 (a)
a = 70 and b = 20
Step-by-step explanation: From the figure in number in number 1(a), we have an isosceles triangle with sides AB and AC being equal in dimensions (as shown by the markings). Therefore angles B and C are also equal (angles subtended by equal sides in an isosceles triangle). If angle B is 3a then angle C equals 3a as well. if the sum of the interior angles of a triangle equals 180, then;
2a+3a+3a = 180
8a = 180
Divide both sides of the equation by 8
a = 22.5
From the figure shown in number 1(b), we have two isosceles triangles. The first on is ABC with sides AC and BC being of equal length and angles A and B being of equal measurement. The second one is triangle ABD with sides AD and BD being of equal length and angles A and B being of equal measurement. In triangle ABC,
A+B+ 64 = 180 (Sum of the interior angles of a triangle)
A+B = 180 - 64
A+B = 116
Remembering that A and B are of equal measurement, we divide 116 by 2 to arrive at angles A and B
Therefore, angle A = 58, and angle B = 58
If angle B equals 58, then in triangle ABD, c is calculated as 58 - 32
Therefore C equals 26.
To calculate angle D (in triangle ABD)
Angle D = 180 - (32+32)
Angle D = 180 - 64
Angle D = 116
Therefore to calculate b;
Remember that the addition of angle D and angle b equals 360 (Sum of angles on a point) Hence we have
D + b = 360
116 + b = 360
Subtract 116 from both sides of the equation
b = 244
From the figure given in number 2(a), we also have two triangles. The first one is isosceles triangle ABD with sides AB and BD being equilateral. Therefore angle A and angle D are equal.
A+B+D = 180 (Sum of the interior angles in a triangle)
A+D+40 = 180
Subtract 40 from both sides of the equation
A+D = 140
Remembering that A and D are equal, divide 140 in two
Angle A = 70, and angle D = 70.
Therefore a = 70
To calculate b, we move on to the second isosceles triangle, triangle BCD with sides BC and BD being equal. That makes angle C and D equal (angles subtended by equal sides of an isosceles triangle)
Angle B measures 140. This is because the line from point D that divides point B gave rise to angle 40 and the other one is 180-40 which equals 140 (Sum of angles on a straight line). If angle B measures 140, and angles C and D are equal, then
C + D = 180 - 140
C + D = 40. We further divide the answer into two
We arrive at C = 20 and D = 20
Therefore b = 20.