Wherever you have the measures of angles mentioned, like, A, B, C, x, y, write them as A deg, x deg with the small circle on top right.
Sum of measures of angles in a triangle = 180 deg. One angle is a right
angle, its measure = 90 deg. Measure of an acute angle is 35 deg.
Measure of the other acute angle = 180 - 90 - 35 = 55 deg
2) Two angles of an Isosceles triangle are equal. Let the measure be A deg. The measure of the third angle is twice A = 2A.
Sum of measures of all the angles is A+ A + 2 A = 180 deg
So 4 A = 180 A = 180 / 4 = 45 deg
So the triangle has two angles of measure 45 deg, and a third angle
measuring 2 * 45 = 90 deg.
measures of the three acute angles of the triangle are : x , x + 12, x - 12
Sum = x + x + 12 + x - 12 = 3 x = 180 deg , as sum of measures is always
180 deg in a triangle.
3 x = 180 deg x = 60 deg
The measures of angles are 60 deg, 60 + 12 = 72 deg, 60 - 12 = 48 deg
a) measure of Exterior angle = sum of measures of Interior angles
105 deg = y + 45 deg So y = 105 - 45 = 60 deg
sum of angles on one side of a straight line = 180 deg = x + 105
So x = 180 - 105 = 75 deg
b) Angle x = angle 75 deg, as the included angles on the opposite side
at A are equal.
sum of measures of angles in triangle ABC = x + y + 40deg = 180
75 + y + 40 = 180 y = 65 deg
c) At A, x = measure of angle BAC, or m BAC = x
At B x = m ABC = x
At C, x = m ACB = x
Sum of measures of angles in triangle ABC, x + x + x = 3 x = 180 deg
So x = 60 deg
At B, x and y are supplementary angles, that is : x + y = 180 deg
So 60 + y = 180 y = 120 deg
The triangle ABC is a right angle triangle with right angle at C. So C = 90.
Sum of angles : x + 40 + 90 = 180 x = 50 deg.
The angle ACB + ACD = 180 AS ACB = 90 deg., ACD = 90 deg
Sum of measures of angles in triangle ACD:
y + 60 + 90 = 180 So y = 30 deg
6) In triangle ABC, the sum of measures of angles is 180 deg.
So 40 + 30 + z = 180 z = 110 deg
The lines XY || BC. So, the included angles on the same side of AYC
are same. That is, y = z So y = 110 deg
Similarly, the included angles on same side of AXB at X and B are same.
That is x = 30 deg
7) The measures of angles are in ratio : 3:4:5
Let us say the measures are 3 x , 4 x and 5 x respectively.
Sum of their measures is 180 deg. So 3x + 4x+ 5x = 180 deg
12 x = 180 x = 15 deg
The angles are 45 deg, 60 deg, 75 deg.
It is an acute angle triangle, as each of the measures is less than 90 deg.
8) measures are in ratio: 1: 2: 6
Let us sy measures are 1 x , 2 x , and 6x respectively.
sum : 1x + 2 x + 6 x = 180 deg So 9 x = 180 deg
x = 20 deg
the measures are : 20 deg, 40 deg, 120 deg.
As one of the angles is an obtuse angle, it is an obtuse angle triangle.