Question
In a triangle ABC, if AB = AC and AB is produced to D such that BD = BC, find $\angle\text{ACD}:\angle\text{ADC}.$
✓
Answer
In the given $\triangle\text{ABC},\text{AB}=\text{AC}$ and $\text{AB}$ is produced to D such that BD = BC
We need to find $\angle\text{ACD}:\angle\text{ADC}$
Now, using the property, "angles opposite to equal sides are equal"
As AB = AC
$\angle6=\angle4\dots({1})$
Similarly,
As AB = AC
$\angle1=\angle4\dots({2})$
Also, using the property, "an exterior angle of the triangle is equal to the sum of the two opposite interior angle"
In $\triangle\text{BDC}$
$\text{ext}.\angle\text{6}=\angle\text{1}+\angle\text{2}$
$\text{ext}.\angle\text{6}=\angle\text{1}+\angle\text{1}(\text{Using 2})$
$\text{ext}.\angle\text{6}=2\angle\text{1}$
From (1), we get
$\angle\text{4}=2\angle\text{1}\dots(3)$
Now, we need to find $\angle\text{ACD}:\angle\text{ADC}$
That is,
$(\angle\text{4}+\angle\text{2}):\angle\text{1}$
$(2\angle\text{1}+\angle\text{2}):\angle\text{1}(\text{Using 3})$
$(2\angle\text{1}+\angle\text{1}):\angle\text{1}(\text{Using 2})$
$3\angle\text{1}:\angle\text{1}$
Eliminating $\angle1$ from both the sides, we get 3:1
Thus, the ratio of $\angle\text{ACD}:\angle\text{ADC} $ is $3 : 1$
We need to find $\angle\text{ACD}:\angle\text{ADC}$
Now, using the property, "angles opposite to equal sides are equal"
As AB = AC
$\angle6=\angle4\dots({1})$
Similarly,
As AB = AC
$\angle1=\angle4\dots({2})$
Also, using the property, "an exterior angle of the triangle is equal to the sum of the two opposite interior angle"
In $\triangle\text{BDC}$
$\text{ext}.\angle\text{6}=\angle\text{1}+\angle\text{2}$
$\text{ext}.\angle\text{6}=\angle\text{1}+\angle\text{1}(\text{Using 2})$
$\text{ext}.\angle\text{6}=2\angle\text{1}$
From (1), we get
$\angle\text{4}=2\angle\text{1}\dots(3)$
Now, we need to find $\angle\text{ACD}:\angle\text{ADC}$
That is,
$(\angle\text{4}+\angle\text{2}):\angle\text{1}$
$(2\angle\text{1}+\angle\text{2}):\angle\text{1}(\text{Using 3})$
$(2\angle\text{1}+\angle\text{1}):\angle\text{1}(\text{Using 2})$
$3\angle\text{1}:\angle\text{1}$
Eliminating $\angle1$ from both the sides, we get 3:1
Thus, the ratio of $\angle\text{ACD}:\angle\text{ADC} $ is $3 : 1$
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