Re: M04, Q7

Question

If quadrilateral  ABCD  is inscribed into a circle, what is the value of  \angle BAD  ?

1.  AC = CD 
2.  \angle ADC = 70^\circ

E 
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Explanation:

If ABCD is a quadrilateral then:

total of all angle's is 360.

Now, in stmt 1 they are saying that AC (which could be the diameter of the circle) = CD (one of the sides of the quad)
- This helps us in understanding that angle ADC = angle DAC = x
 and that x + x + angle ACD = 180
 and that angle ACD = 180 - 2x
 What we need is angel BAD which is not provided in Stmt 1, thus NS

Stmt 2 
gives angel ADC = 70 deg
which does not help us in estimating any other angle since a quadrilateral can be a square, parallelogram or a rhombus so angles can be different

Combining stmt 1 and stmt 2 we get
angle ADC = 70 deg = angle CAD
But we are still missing information about the other angle's i.e.
angle ABC + angle BCD + angle ADC + angle BAD = 360
angle ABC + angle BCD + angle 70 + (angle BAC + angle CAD) = 360
angle ABC + angle BCD + angle 70 + (angle BAC + 70) = 360

Rest of the angles above are still missing. Thus, E
If we were told what kind of a quad it is then it would be easier to find out the angle.

nightwing79 · Answer

Well explained - thank you.

+1

https://www.algebralab.org/lessons/lesso ... istics.xml

I think it is important to remember a quadrilateral can be a shape of four sides. Any shape is fine. A warped Trapezium of obtuse angles would be a qdrltlt as well. Am I right?

adhithya · Answer

dude,ac cannot be diameter of circle because if that were true,adc=90 deg but we cant be sure of it

druid007 · Answer

since ADC = 70 and AC = CD,

DAC = 70 and therefore ACD = 180 - 140 = 40

since inscribed angle of point on circle to A and D is 40,center of circle = O

AOD = 2 * 40 = 80
in triangle AOD, OAD = ODA = 40

BOC = AOD = 80
therefore AOB = DOC = (360 - 80 -80) / 2 = 100

in triangle AOB, OAB = OBA = 50

BAD = OAB + OAD = 40 + 50 = 90

Answer = C (both statements together)

kingfisher · Answer

hey how can we say that centre lies on the dia AC.

adhithya · Answer

how is ac the diameter

EkaterinaZ · Answer

The answer is E.

We can just find out all angles of ACD - CAD (70), CDA (70), DCA (40). However, we cannot find any other angle.

drozzy · Answer

I know the answer.... its E!

Explanation:
I banged my head against the wall after wasting 15 minutes on this question.
I then figured out that I still cannot get to the answer.
So it has to be E.

PS: I think that was wasteful seriously.

drozzy · Answer

my question as well...

before having a kudos/+1 party please answer the eternal question... "how is ac the diameter?"

I had imagined the quadrilateral ABCD inscribed in the the circle at the very top.... let's say in a semi-circle. now how does AC become the diameter???

321kumarsushant · Answer

Hey dude!
you cant assume AC as the diameter of circle.
even if you have assumed this, then the quad will be a Rectangle or Square and its very easy to find the angles with these 2 stmnts.

Sol:
Stmt 2: Angl ADC = 70
stmt 1: AC=DC i.e. angl CAD= angl CDA =70
==> angl DCA =40.
since, Angl ADC + Angl ABC =180; 
==> Angl ABC = 110.

we cant find Angl BAD = 70 + Angl BAC, using these 2 condition.
so FInal ANs is E.

AndreG · Answer

321, how do you get that ABC =110?

We know all angles in the the small triangle ACD. But now B could be ANY point above line AC on the cirlce - does it always end up as the same angle?? If so, please elaborate, how to get that angle.

Regarding the question if AC could or could not be the diameter:
Statement 2 tells you that it is NOT!
Because if AC was the diameter, then ABC and ADC would both be right angles. (but S2 tells you ADC = 70)

321kumarsushant · Answer

Hey, Theorem : The sum of the opposite angles in cyclic Quadrilateral are supplementary to one another, i.e., their sum is equal to two right angles so angl ADC + Angl ABC = 180, n angl ADC = 70. its the property of cyclic quad. for more theorem follow the link. https://www.tutornext.com/cyclic-quadril ... rties/1028

anandthiru · Answer

S1: isosceles triangle, so Angle ADC=DAC. Not sufficient
S2: Angle ADC=70 deg. Not sufficient.
Combining we have Angle ACD=40 (Sum of 3 angles in a triangle ADC=180 Deg). 
Now that alternate opp. Angles are equal AC bisects AB & DC, we can say Angle CAB=40. 
Therefore Angle BAD=110 deg.

kudos if you like my explanation !

vittarr · Answer

it is not statex thar it is paralellogram so ur explanation os wrong
E

Posted from my mobile device

Jdam · Answer

I'm glad there are so many opinions on this answer

I got E. I can only get the measurement for half of angle DAB.

AzWildcat1 · Answer

I drew ABCD as a rectangle with the points on the circle. I ended up with answer D and was surprised to find out it was E.

Since Rectangles have 4 rt angles, for statement 1 I drew a diagonal line through the rectangle to form 2 rt triangles, and since ac = cd, I thought it was isosceles thus I had <ADC and < DAC = 45 each therefore able to answer the original question.

For statement 2, Since <ADC=70, I was able to use a similar method and turned the rectangle into 2 rt triangles. I added 90 and 70 and then subtracted from 180 to find <DAC. I then was able to find out the value for <BAD by subtracting <DAC from 90, thus I labeled statement 2 as sufficient.
 I was wondering what I did wrong since I was off by a lot.

teal · Answer

A quick question:

For which quadrilaterals do diagonals bisect the vertex angles?

Re: M04, Q7

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