Question

The two circles intersects at C and D, where O is the centre of the second circle. AD produced and cuts the second circle at F. BD produced and cuts the second circle at E. $\angle$DEF $={110}^o$, $\angle$ACB $={32}^o$ and $\angle$DAB $={118}^o$. Find (ii) $\angle$ACE, (ii) $\angle$COD.

• A. (i), ${96}^o$, (ii) ${68}^o$
• B. (i) ${72}^o$, (ii) ${59}^o$
• C. (i) ${59}^o$, (ii) ${72}^o$
• D. (i) ${68}^o$, (ii) ${96}^o$

Answer 1

Answer: (D)

(i) ${68}^o$, (ii) ${96}^o$

$\angle DEF={110}^o\angle ACB={32}^o\angle DAB={118}^o$

$ABCD$ and $CDEF$ are cyclic quadrilateral

$\angle BAD+\angle BCD={180}^{o}118+\angle BCD=180\angle BCD=62\angle BCA+\angle ACD=6232+\angle 62\angle ACD=30$

In $ABCD$ cyclic quadrilateral

$\angle BCA=\angle BDA$

$\angle BDA={32}^o$ (angle in major sector from same chord)

$\angle BDA=\angle EDF={32}^o$ (opposite angle)

In $△DEF$

$\angle DEF+\angle EDF+\angle DFE=180\angle DFE=180-32-110={38}^o$

$\angle DFE=\angle DCE={38}^o$ (angle in major segment from same chord)

$i)\angle ACE=\angle ACD+\angle DCE=30+38={68}^o$

$ii)\angle COD={96}^o$