Question

(a) Find the value of x in the below diagram given that, $\angle ACB$ = 25${}^o$, $\angle CBA$ = 15${}^o$ and CD is the height of C from the extended line segment BA:




(b) The lengths of two sides of a triangle are 13 cm and 16 cm. The third side should lie between 'a' cm and 'b' cm for the triangle to be formed. Find the value of a + b? [4 MARKS]

Answer 1

Each part: 2 Marks

In the given triangle ABC,
(a) Sum of all angles of a triangle = 180${}^o$
Or, $\angle CAB$ + $\angle ABC$ + $\angle BCA$ = 180${}^o$
Or, $\angle CAB$ + 15${}^o$ + 25${}^o$ = 180${}^o$
Or, $\angle CAB$ = 180${}^o$ - 40${}^o$ = 140${}^o$

For the triangle ACD,
$\angle D$ + X = $\angle CAB$ ($\because$ sum of interior opposite angles = exterior angle)
90${}^o$ + X = 140${}^o$ ($\because$ CD is an altitude from C to extended BA, $\therefore$ $\angle D$ = 90${}^o$)
Or, X = 50${}^o$


(b) The third side of a triangle must be greater than the difference between the other two sides.

That is, third side $>$ (16 - 13) which is 3 cm.

Also, Sum of lengths of any two sides of a triangle is always greater than the third side.

i.e., third side $<$ (16+13) which is 29 cm.

Hence, a + b = 3 + 29 = 32.