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Sub 505 (Easy)|   Geometry|      
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HarveyKlaus
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In the figure below, if AB = BD and the measure of angle BDC is 102 Updated on: 23 May 2017, 06:37
Originally posted by HarveyKlaus on 10 Jul 2016, 02:56.
Last edited by Bunuel on 23 May 2017, 06:37, edited 2 times in total.
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In the figure below, if AB = BD and the measure of angle BDC is 102, what is the measure, in degrees, of the angle ABD?
Attachment:
Screen Shot 2016-07-10 at 11.54.42.png
Screen Shot 2016-07-10 at 11.54.42.png [ 33.62 KiB | Viewed 18860 times ]


A) 24
B) 39
C) 51
D) 60
E) can't be determined

Show Spoiler
As we know that our triangle ABD is an isosceles triangle, so two angles are going to be equal. Now there are 2 options, either two of the angles are 78 degrees and the third one is 29 or one is 78 degrees and the other two are of 51. Im not sure how to to choose between these cases?

I appreciate your help.

Thanks
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A
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rohit8865
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In the figure below, if AB = BD and the measure of angle BDC is 102 10 Jul 2016, 04:19
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HarveyKlaus
In the figure below, if AB=BD and the measure of angleBCD is 102, what is the measure, in degrees, of the angle ABD?

A) 24
B) 39
C) 51
D) 60
E) can't be determined

As we know that our triangle ABD is an isosceles triangle, so two angles are going to be equal. Now there are 2 options, either two of the angles are 78 degrees and the third one is 29 or one is 78 degrees and the other two are of 51. Im not sure how to to choose between these cases?

I appreciate your help.

Thanks
Show more

I hope its angle BDC=102Deg.
Anyway corrected fig. is given

Angle created by legs of isosceles triangle are equal
Here AB and AC are equal so angle related to them BAC and ACB are equal..
Coming to answer part
S0 angle BCA+BCD=180
BCA=72
AngleBCA=angle BAC=78Deg. (isosceles triangle)
Sum of angles of triangle=ABC=180
Angle ABC=24
Ans A
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In the figure below, if AB = BD and the measure of angle BDC is 102 Updated on: 10 Jul 2016, 05:21
Originally posted by HarveyKlaus on 10 Jul 2016, 04:27.
Last edited by HarveyKlaus on 10 Jul 2016, 05:21, edited 1 time in total.
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Hi Rohit,

Thanks for the message. Sorry yea its angle BDC - I have corrected it in the original question. But that still does not answer the argument I have raised in the question. What am I missing?


rohit8865
HarveyKlaus
In the figure below, if AB=BD and the measure of angleBCD is 102, what is the measure, in degrees, of the angle ABD?

A) 24
B) 39
C) 51
D) 60
E) can't be determined

As we know that our triangle ABD is an isosceles triangle, so two angles are going to be equal. Now there are 2 options, either two of the angles are 78 degrees and the third one is 29 or one is 78 degrees and the other two are of 51. Im not sure how to to choose between these cases?

I appreciate your help.

Thanks

I hope its angle BDC=102Deg.
Anyway corrected fig. is given

Angle created by legs of isosceles triangle are equal
Here AB and AC are equal so angle related to them BAC and ACB are equal..
Coming to answer part
S0 angle BCA+BCD=180
BCA=72
AngleBCA=angle BAC=78Deg. (isosceles triangle)
Sum of angles of triangle=ABC=180
Angle ABC=24
Ans A
Show more
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In the figure below, if AB = BD and the measure of angle BDC is 102 10 Jul 2016, 04:44
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[quote="HarveyKlaus"]In the figure below, if AB=BD and the measure of angle BDC is 102, what is the measure, in degrees, of the angle ABD?

A) 24
B) 39
C) 51
D) 60
E) can't be determined

As we know that our triangle ABD is an isosceles triangle, so two angles are going to be equal. Now there are 2 options, either two of the angles are 78 degrees and the third one is 29 or one is 78 degrees and the other two are of 51. Im not sure how to to choose between these cases?

I appreciate your help.

Hi,

This is how I tried and I got it correct.

Given ABD is isosceles triangle i..e AB = BD i.e. angles BAD = BDA. ( for isosceles triangle two triangle will be equal and the third one will be different)

NOTE : The same two angles can be greater than or lesser than the third angle and this is the main point.

Given angle BDC = 102 degrees.

Then remember one theorem that external side of triangle is sum of two interior angles.

i.e. 102 = angle B + angle A ---> eq 1.

Here angle A = angle D ( from isosceles triangle ABD)

We also know that sum of three angles in a triangle is 180 i.e. angle A + angle D + + angle B = 180 ( from isosceles triangle ABD).

i.e. two angles of A or D + angle B = 180 = > 2 A + B = 180. ( note A = D as they are same).

From options take B as 24 and sub in eq 1.

102 = angle B + angle A.

angle A = 102 - 24 = 78.

When angle A = 78 = angle D.

Then sum of angles A + D + B = 78 + 78 + 24 = 180. I substituted the angles from options E to A but took 3 mins to solve.

Correct option is A.
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In the figure below, if AB = BD and the measure of angle BDC is 102 10 Jul 2016, 04:58
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msk0657
HarveyKlaus
In the figure below, if AB=BD and the measure of angle BDC is 102, what is the measure, in degrees, of the angle ABD?

A) 24
B) 39
C) 51
D) 60
E) can't be determined

As we know that our triangle ABD is an isosceles triangle, so two angles are going to be equal. Now there are 2 options, either two of the angles are 78 degrees and the third one is 29 or one is 78 degrees and the other two are of 51. Im not sure how to to choose between these cases?

I appreciate your help.

Hi,

This is how I tried and I got it correct.

Given ABD is isosceles triangle i..e AB = BD i.e. angles BAD = BDA. ( for isosceles triangle two triangle will be equal and the third one will be different)

NOTE : The same two angles can be greater than or lesser than the third angle and this is the main point.

Given angle BDC = 102 degrees.

Then remember one theorem that external side of triangle is sum of two interior angles.

i.e. 102 = angle B + angle A ---> eq 1.

Here angle A = angle D ( from isosceles triangle ABD)

We also know that sum of three angles in a triangle is 180 i.e. angle A + angle D + + angle B = 180 ( from isosceles triangle ABD).

i.e. two angles of A or D + angle B = 180 = > 2 A + B = 180. ( note A = D as they are same).

From options take B as 24 and sub in eq 1.

102 = angle B + angle A.

angle A = 102 - 24 = 78.

When angle A = 78 = angle D.

Then sum of angles A + D + B = 78 + 78 + 24 = 180. I substituted the angles from options E to A but took 3 mins to solve.

Correct option is A.
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I think u need to frame question properly its AB=BC if AB=BD then sum of angles (triangle ABD)were > 180

Thanks
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In the figure below, if AB = BD and the measure of angle BDC is 102 23 May 2017, 02:44
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HarveyKlaus
In the figure below, if AB=BD and the measure of angle BDC is 102, what is the measure, in degrees, of the angle ABD?

A) 24
B) 39
C) 51
D) 60
E) can't be determined

As we know that our triangle ABD is an isosceles triangle, so two angles are going to be equal. Now there are 2 options, either two of the angles are 78 degrees and the third one is 29 or one is 78 degrees and the other two are of 51. Im not sure how to to choose between these cases?

I appreciate your help.

Thanks
Show more


Hi!
Can I ask Bunuel to answer this question please? Would greatly appreciate!
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In the figure below, if AB = BD and the measure of angle BDC is 102 23 May 2017, 06:51
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michaelkalend
HarveyKlaus
In the figure below, if AB=BD and the measure of angle BDC is 102, what is the measure, in degrees, of the angle ABD?

A) 24
B) 39
C) 51
D) 60
E) can't be determined

As we know that our triangle ABD is an isosceles triangle, so two angles are going to be equal. Now there are 2 options, either two of the angles are 78 degrees and the third one is 29 or one is 78 degrees and the other two are of 51. Im not sure how to to choose between these cases?

I appreciate your help.

Thanks

Hi!
Can I ask Bunuel to answer this question please? Would greatly appreciate!
Show more

In the figure below, if AB = BD and the measure of angle BDC is 102, what is the measure, in degrees, of the angle ABD?

A) 24
B) 39
C) 51
D) 60
E) can't be determined


Check below:


Angles ADB and CDB add up to 180°, thus (angle ADB) = 180° - (angle CDB) = 180° - 102° = 78°.

Since AB = BD, then ABD is an isosceles triangle and the angles at the base are equal. (angle ADB) = (angle DAB) = 78°.

All three angles in a triangle add up to 180°, thus (angle ADB) + (angle DAB) + (angle ABD) = 78° + 78° + (angle ABD) = 180°.

Solving gives (angle ABD) =24°.

Answer: A.

Hope it's clear.

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In the figure below, if AB = BD and the measure of angle BDC is 102 23 May 2017, 07:24
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Bunuel
michaelkalend
HarveyKlaus
In the figure below, if AB=BD and the measure of angle BDC is 102, what is the measure, in degrees, of the angle ABD?

A) 24
B) 39
C) 51
D) 60
E) can't be determined

As we know that our triangle ABD is an isosceles triangle, so two angles are going to be equal. Now there are 2 options, either two of the angles are 78 degrees and the third one is 29 or one is 78 degrees and the other two are of 51. Im not sure how to to choose between these cases?

I appreciate your help.

Thanks

Hi!
Can I ask Bunuel to answer this question please? Would greatly appreciate!

In the figure below, if AB = BD and the measure of angle BDC is 102, what is the measure, in degrees, of the angle ABD?

A) 24
B) 39
C) 51
D) 60
E) can't be determined


Check below:


Angles ADB and CDB add up to 180°, thus (angle ADB) = 180° - (angle CDB) = 180° - 102° = 78°.

Since AB = BD, then ABD is an isosceles triangle and the angles at the base are equal. (angle ADB) = (angle DAB) = 78°.

All three angles in a triangle add up to 180°, thus (angle ADB) + (angle DAB) + (angle ABD) = 78° + 78° + (angle ABD) = 180°.

Solving gives (angle ABD) =24°.

Answer: A.

Hope it's clear.

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Very clear and became so much easy! Great! Thanks a lot!
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In the figure below, if AB = BD and the measure of angle BDC is 102 23 May 2017, 08:26
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HarveyKlaus
In the figure below, if AB = BD and the measure of angle BDC is 102, what is the measure, in degrees, of the angle ABD?
Attachment:
Screen Shot 2016-07-10 at 11.54.42.png


A) 24
B) 39
C) 51
D) 60
E) can't be determined
Show more

∠BDC + ∠BDA = 180°

Or, 102° + ∠BDA = 180°

So, ∠BDA = 78°

Now, in ΔABD , ∠BDA = ∠BAD

So, we have ∠BAD + ∠BDA + ∠ABD = 180°

Or, 78° + 78° + ∠ABD = 180°

Or, ∠ABD = 24°

Thus, answer will be (A) 24°
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In the figure below, if AB = BD and the measure of angle BDC is 102 25 Jul 2025, 05:07
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