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This question is from a family friend's 11th grade geometry homework:

If $BC$ is the bisector of the angle $A\hat{B}D$ , use the following information to determine the missing values: Measurment of angle $\hat{ABD}= 7x+9 $, measurement of angle $\hat{CBD} = 3x+5$ , $x=$ ?. Also determine the values for the measurements of angles $\hat{ABC}, \hat{CBD} $, and $\hat{ABD}$ .

I may be overthinking it, but is this question flawed? There is no other information that would allow for the application of the Angle Bisector Theorem, unless I am misunderstanding.

dmtri
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DaveAlex1120
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  • By the Theorem, you have $3x+5=7x+9$. What else do you need? – Rushabh Mehta Oct 03 '19 at 14:25
  • If point C bisects segment AD then I don't see how this would work when drawn out. angle CBD is created by the bisecting segment BC. Angle ABD would be the sum of angles ABC and CBD. – DaveAlex1120 Oct 03 '19 at 14:29
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    @DaveAlex1120: An angle bisector bisects the angle, not the opposite side. (Well, it can do both in a special circumstance.) The line through a vertex and the midpoint of the opposite side is a median. – Blue Oct 03 '19 at 14:38
  • Don apparently thought AD was the bisector. The answer below does not make that mistake. – David K Oct 03 '19 at 21:21

3 Answers3

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$\angle ABD=2\angle CBD$ or $7x+9=2(3x+5)$. This yields $x=1$. Then $\angle ABD=7\cdot1+9=16$ and $\angle ABC=\angle CBD=8$

Andrei
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$\angle ABD = 7x + 9$

$\angle CBD = 3x + 5$

Since BC is the angle bisector of $\angle ABD$, we have: $\angle ABC = \angle CBD$

Also, $\angle ABD$

$$= \angle ABC + \angle CBD$$

$$= \angle CBD + \angle CBD$$

$$= 2 \angle CBD$$

Thus we have $7x + 9 = 2 (3x + 5)$.

On solving this, you'll get $x = 1$.

So we have $\angle(ABD) = 7*1 + 9 = 16 $degrees

$\angle CBD = 3*1 + 5 = 8$ degrees

$\angle ABC = \angle CBD = 8 $degrees

Arcchit Jain
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It is very simple since $BC$ is bisector of $\angle ABD $ so It's twice as much as $\angle CBD $ . So $$7x+9=2(6x+10)$$ or $$x=1$$. Can you can calculate rest ?

Rishi
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  • I am glad it cleared your confusion . Now accept any of one answer so that it doesn't remain in unanswered category . – Rishi Oct 03 '19 at 14:57