MHB Angle RAB in Triangle PQR: 84°, 78°, 48°, 63°

  • Thread starter Thread starter anemone
  • Start date Start date
  • Tags Tags
    Angle
anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
In a triangle $PQR$, $\angle P=84^{\circ}$, $\angle R=78^{\circ}$. Points $A$ and $B$ are on $PQ$ and $QR$ so that $\angle PRA=48^{\circ}$ and $\angle RPE=63^{\circ}$.

What is the measure of $\angle RAB$?
 
Mathematics news on Phys.org
anemone said:
In a triangle $PQR$, $\angle P=84^{\circ}$, $\angle R=78^{\circ}$. Points $A$ and $B$ are on $PQ$ and $QR$ so that $\angle PRA=48^{\circ}$ and $\angle RPB=63^{\circ}$.

What is the measure of $\angle RAB$?

My solution:
 

Attachments

  • Angle RAB.jpg
    Angle RAB.jpg
    18 KB · Views: 130
Last edited by a moderator:
I couldn't quite follow Albert's solution.

The following also uses the circumcircle of triangle RAB. Also it's written as a slight generalization. It is critical that angle ARQ is 30 degrees. The original problem has $\theta$ equal to 21. So angle RAB is 81.
sl4dqx.png
 
Last edited by a moderator:

Attachments

  • Angle of RAB.jpg
    Angle of RAB.jpg
    18.7 KB · Views: 106
Last edited:
Albert,
I understand perfectly if quadrilateral PDAB is cyclic. This was my original problem, and I still can't see why it's true.
 
johng said:
Albert,
I understand perfectly if quadrilateral PDAB is cyclic. This was my original problem, and I still can't see why it's true.
RDAB is cyclic not PDAB
 
Albert,
Sorry, it was a typo. My problem is: why is quadrilateral RDAB cycliC? I can prove this only by knowing the answer.
 
johng said:
I couldn't quite follow Albert's solution.

The following also uses the circumcircle of triangle RAB. Also it's written as a slight generalization. It is critical that angle ARQ is 30 degrees. The original problem has $\theta$ equal to 21. So angle RAB is 81.

Thanks, johng for participating and your solution is correct, well done!:)
Albert said:
cyclic-quadrilateral-and-its-properties:
Cyclic Quadrilateral and its Properties | TutorNext.com#

Hi Albert,

First, thanks for participating in this challenge problem of mine.:)

But, looking at your solution, if you mean to verify that the specific quadrilateral RDAB is a cyclic quadrilateral by first showing the sum of the opposite angles in it are supplementary, then I am unable to follow it.:confused:
 
  • #10
Albert said:
johng:
Can you follow my solution now ?
anemone:
If you are still unable to follow it,I will use the following method (forget the circle)
$x=\angle DRA=30^o=\angle ARB$
$\angle ABQ=x+y=111^o=63+18+x=63^o+48^o$
$\angle RAB=111-30=81^o=y$
 
Last edited:

Similar threads

Replies
4
Views
2K
Replies
2
Views
2K
Replies
2
Views
2K
Replies
17
Views
5K
Replies
13
Views
3K
Replies
2
Views
2K
Replies
2
Views
2K
Replies
3
Views
2K
Back
Top