You visited us 1 times! Enjoying our articles? Unlock Full Access!

Properties of Cyclic Quadrilateral

In ABC seg AB...

Question

In $△$ ABC seg AB = seg AC and P is any point on AC. Through C, a line is drawn to intersect BP produced in Q such that $∠$ ABQ = $∠$ ACQ. Prove that $∠ ACQ = 90 ° + \frac{1}{2} ∠ BAC$

Open in App

Solution

Disclaimer :- It should be $∠ AQC instead of ∠ ACQ$ in the question.

$In △ ABC AB = AC So, ∠ ABC = ∠ ACB (angles opposite to equal sides) ∠ ABQ = ∠ ACQ . . . . . . . . (1) Since, seg AQ subtends equal angles at points B and C which are on the same side of line AQ . So, A, B, C and Q are concyclic i . e . ABCQ is a cyclic quadrilateral . Now, ∠ AQC + ∠ ABC = 180 ° (ABCQ is a cyclic quadrilateral) \Rightarrow ∠ ABC = 180 ° - ∠ AQC . . . . . . . . . . . (2) In △ ABC ∠ ABC + ∠ ACB + ∠ BAC = 180 ° (angles sum property) \Rightarrow 2 ∠ ABC + ∠ BAC = 180 ° \Rightarrow 2 ∠ ABC = 180 ° - ∠ BAC \Rightarrow ∠ ABC = 90 ° - \frac{1}{2} ∠ BAC \Rightarrow 180 ° - ∠ AQC = 90 ° - \frac{1}{2} ∠ BAC (using eqn 1) \Rightarrow ∠ AQC = 180 ° - 90 ° + \frac{1}{2} ∠ BAC ∠ AQC = 90 ° + \frac{1}{2} ∠ BAC$

Suggest Corrections

Similar questions

A B C

is an isosceles triangle in which

A B = A C

P

is any point in the interior of

Δ A B C

such that

∠ A B P = ∠ A C P

. Prove that
a)

B P = C P

A P

bisects

∠ B A C

Q. In

△ A B C

P Q

is a line segment intersecting

A B

P

and

A C

Q

such that

s e g P Q ∥ s e g B C

. If

P Q

divides

△ A B C

into two equal parts means equal in area, find

\frac{B P}{A B}

Q. In the figure,seg

A C

and seg

B D

intersect each other in point

P

such that

\frac{A P}{C P} = \frac{B P}{D P}

.Prove that,

△ A B P \sim △ C D P

The side $A B$ of a parallelogram $A B C D$ is produced to any point $P$ . A line through $A$ and parallel to $C P$ meets $C B$ produced at $Q$ and then parallelogram $P B Q R$ is completed (see Fig. 9.26). Show that ar $(A B C D)$ $=$ ar $(P B Q R)$ .

Q. In

U A B C^e B A C = 90^{o}, A B = A C s e g A P o

side

B C, B - P - C

D

is any points on side

B C

. Prove that:

2 A D^{2} = B D^{2} + C D^{2}

Join BYJU'S Learning Program

In △ABC seg AB = seg AC and P is any point on AC. Through C, a line is drawn to intersect BP produced in Q such that ∠ABQ = ∠ACQ. Prove that ∠ACQ=90°+12∠BAC

In $△$ ABC seg AB = seg AC and P is any point on AC. Through C, a line is drawn to intersect BP produced in Q such that $∠$ ABQ = $∠$ ACQ. Prove that $∠ ACQ = 90 ° + \frac{1}{2} ∠ BAC$