Question

PLEASE HELP!!!

Let $ABCD$ be a cyclic quadrilateral. Let $P$ be the intersection of $\overline{AD}$ and $\overline{BC}$, and let $Q$ be the intersection of $\overline{AB}$ and $\overline{CD}$. Prove that the angle bisectors of $\angle DPC$ and $\angle AQD$ are perpendicular.

Answer 1

Let \(ABCD\) be a cyclic quadrilateral.
Let \(P\) be the intersection of \(\overline{AD}\)
and \(\overline{BC}\), and let \(Q\) be the intersection
of \(\overline{AB}\) and \(\overline{CD}\). Prove that the
angle bisectors of \(\angle DPC\) and \(\angle AQD\) are perpendicular.
question

Answer 2

still here?

Answer 3

yup

Answer 4

you must be knowing that opposite angles in a cyclic quadrilateral add up to 180

Answer 5

ohhh i see

Answer 6

As a start, use that to show that \(\angle DAB \cong \angle BCQ\)

Answer 7

oh i see where you're going! thanks so much! I'll try it out myself first and come back if i need more help!

Answer 8

**
1) By triangle exterior angle theorem, \(\angle DAB\cong \angle P + \angle B\).

2) Since the opposite angles in a cyclic quadrilateral add up to \(180\) :
\(\angle DAB\cong \angle BCQ\)

3) From above two steps, we get
\(\angle BQC \cong 180-(\angle P+\angle B) - \angle B\)

Answer 9

Okay, I'll let you finish it off.. good luck!

Answer 10

for what it is worth, here is a complicated approach