Answers

2014-08-27T19:12:15+05:30
1) 
One of the acute angles = 35 degrees
The measure of the third angle = 90 - 35 = 55 degrees

2) Let the equal angle measure x degree each
Third angle = 2x degrees
Sum of three angles = 180 degrees

Thats is x + x + 2x = 180

4x = 180

x = 45 degrees

Two angles are 45 degrees each and the third angle is 90 degrees.

3)
sum of the three angles of a triangle is 180 degrees

x + x + 12 + x - 12 = 180 degree
3x = 180
x = 60







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2014-08-27T20:33:26+05:30

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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.

1) 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.

3 )
       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

4 )
   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

5)
     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.

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