Compute the angle in the quadrilateral
Question: Let $ABCD$ be a convex quadrilateral with $∠DAC = ∠ACD = 17^{circ}$; $angle CAB = 30^{circ}$; and $angle BCA = 43^{circ}$. Compute $angle ABD$.
What I have so far:
Since $angle DAC = angle ACD = 17^{circ}$ , points $A$,$C$, and $D$ form an isosceles triangle. We can look at the circle with center $D$ and radius $|DA|=|DC|$. I believe this can help because one can then look at the angles subtended by the arc AC and use them to gain further information (I have seen this done in a similar problem). I can't seem to find any angles that are $frac{146}2=73^{circ},$ though so this is probably wrong.
Any help is appreciated
geometry euclidean-geometry
edited Dec 12 '18 at 11:29
amWhy
192k28225439
asked Dec 12 '18 at 8:01
ricorico
806
add a comment |
Question: Let $ABCD$ be a convex quadrilateral with $∠DAC = ∠ACD = 17^{circ}$; $angle CAB = 30^{circ}$; and $angle BCA = 43^{circ}$. Compute $angle ABD$.
What I have so far:
Since $angle DAC = angle ACD = 17^{circ}$ , points $A$,$C$, and $D$ form an isosceles triangle. We can look at the circle with center $D$ and radius $|DA|=|DC|$. I believe this can help because one can then look at the angles subtended by the arc AC and use them to gain further information (I have seen this done in a similar problem). I can't seem to find any angles that are $frac{146}2=73^{circ},$ though so this is probably wrong.
Any help is appreciated
geometry euclidean-geometry
edited Dec 12 '18 at 11:29
amWhy
192k28225439
asked Dec 12 '18 at 8:01
ricorico
806
Is this a circular quadrilateral (has a circumcircle?)
– user376343
Dec 12 '18 at 16:31
add a comment |
Question: Let $ABCD$ be a convex quadrilateral with $∠DAC = ∠ACD = 17^{circ}$; $angle CAB = 30^{circ}$; and $angle BCA = 43^{circ}$. Compute $angle ABD$.
What I have so far:
Since $angle DAC = angle ACD = 17^{circ}$ , points $A$,$C$, and $D$ form an isosceles triangle. We can look at the circle with center $D$ and radius $|DA|=|DC|$. I believe this can help because one can then look at the angles subtended by the arc AC and use them to gain further information (I have seen this done in a similar problem). I can't seem to find any angles that are $frac{146}2=73^{circ},$ though so this is probably wrong.
Any help is appreciated
geometry euclidean-geometry
edited Dec 12 '18 at 11:29
amWhy
192k28225439
asked Dec 12 '18 at 8:01
ricorico
806
Question: Let $ABCD$ be a convex quadrilateral with $∠DAC = ∠ACD = 17^{circ}$; $angle CAB = 30^{circ}$; and $angle BCA = 43^{circ}$. Compute $angle ABD$.
What I have so far:
Since $angle DAC = angle ACD = 17^{circ}$ , points $A$,$C$, and $D$ form an isosceles triangle. We can look at the circle with center $D$ and radius $|DA|=|DC|$. I believe this can help because one can then look at the angles subtended by the arc AC and use them to gain further information (I have seen this done in a similar problem). I can't seem to find any angles that are $frac{146}2=73^{circ},$ though so this is probably wrong.
Any help is appreciated
geometry euclidean-geometry
geometry euclidean-geometry
edited Dec 12 '18 at 11:29
amWhy
192k28225439
asked Dec 12 '18 at 8:01
ricorico
806
edited Dec 12 '18 at 11:29
amWhy
192k28225439
asked Dec 12 '18 at 8:01
ricorico
806
edited Dec 12 '18 at 11:29
amWhy
192k28225439
edited Dec 12 '18 at 11:29
amWhy
192k28225439
edited Dec 12 '18 at 11:29
amWhy
192k28225439
192k28225439
asked Dec 12 '18 at 8:01
ricorico
806
asked Dec 12 '18 at 8:01
ricorico
806
asked Dec 12 '18 at 8:01
ricorico
806
806
Is this a circular quadrilateral (has a circumcircle?)
– user376343
Dec 12 '18 at 16:31
add a comment |
Is this a circular quadrilateral (has a circumcircle?)
– user376343
Dec 12 '18 at 16:31
Is this a circular quadrilateral (has a circumcircle?)
– user376343
Dec 12 '18 at 16:31
Is this a circular quadrilateral (has a circumcircle?)
– user376343
Dec 12 '18 at 16:31
add a comment |
1 Answer
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reflex $angle ADC=360^{circ}-146^{circ}=214^{circ}=2times 107^{circ}=2times angle ABC$
Since the angle subtended at the center of a circle is double the angle subtended at the circumference, above proves that our constructed circle also passes through the point $B$. So, $|DA|=|DB|=|DC|$ which in turn means that $angle ABD=angle BAD=angle DAC+angle BAC=47^{circ}$.
answered Dec 12 '18 at 11:22
Sameer BahetiSameer Baheti
5568
Did you pay attention at "reflex"? According to the question which I think I've solved correctly, $ABCD$ is a quadrilateral whose vertex $D$ can be thought of as lying on the center of a circle with the other three lying on the circumference.
– Sameer Baheti
Dec 12 '18 at 16:36
add a comment |
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1 Answer
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reflex $angle ADC=360^{circ}-146^{circ}=214^{circ}=2times 107^{circ}=2times angle ABC$
Since the angle subtended at the center of a circle is double the angle subtended at the circumference, above proves that our constructed circle also passes through the point $B$. So, $|DA|=|DB|=|DC|$ which in turn means that $angle ABD=angle BAD=angle DAC+angle BAC=47^{circ}$.
answered Dec 12 '18 at 11:22
Sameer BahetiSameer Baheti
5568
Did you pay attention at "reflex"? According to the question which I think I've solved correctly, $ABCD$ is a quadrilateral whose vertex $D$ can be thought of as lying on the center of a circle with the other three lying on the circumference.
– Sameer Baheti
Dec 12 '18 at 16:36
add a comment |
reflex $angle ADC=360^{circ}-146^{circ}=214^{circ}=2times 107^{circ}=2times angle ABC$
Since the angle subtended at the center of a circle is double the angle subtended at the circumference, above proves that our constructed circle also passes through the point $B$. So, $|DA|=|DB|=|DC|$ which in turn means that $angle ABD=angle BAD=angle DAC+angle BAC=47^{circ}$.
answered Dec 12 '18 at 11:22
Sameer BahetiSameer Baheti
5568
Did you pay attention at "reflex"? According to the question which I think I've solved correctly, $ABCD$ is a quadrilateral whose vertex $D$ can be thought of as lying on the center of a circle with the other three lying on the circumference.
– Sameer Baheti
Dec 12 '18 at 16:36
add a comment |
reflex $angle ADC=360^{circ}-146^{circ}=214^{circ}=2times 107^{circ}=2times angle ABC$
Since the angle subtended at the center of a circle is double the angle subtended at the circumference, above proves that our constructed circle also passes through the point $B$. So, $|DA|=|DB|=|DC|$ which in turn means that $angle ABD=angle BAD=angle DAC+angle BAC=47^{circ}$.
answered Dec 12 '18 at 11:22
Sameer BahetiSameer Baheti
5568
reflex $angle ADC=360^{circ}-146^{circ}=214^{circ}=2times 107^{circ}=2times angle ABC$
Since the angle subtended at the center of a circle is double the angle subtended at the circumference, above proves that our constructed circle also passes through the point $B$. So, $|DA|=|DB|=|DC|$ which in turn means that $angle ABD=angle BAD=angle DAC+angle BAC=47^{circ}$.
answered Dec 12 '18 at 11:22
Sameer BahetiSameer Baheti
5568
edited Dec 12 '18 at 11:40
answered Dec 12 '18 at 11:22
Sameer BahetiSameer Baheti
5568
answered Dec 12 '18 at 11:22
Sameer BahetiSameer Baheti
5568
answered Dec 12 '18 at 11:22
Sameer BahetiSameer Baheti
5568
5568
Did you pay attention at "reflex"? According to the question which I think I've solved correctly, $ABCD$ is a quadrilateral whose vertex $D$ can be thought of as lying on the center of a circle with the other three lying on the circumference.
– Sameer Baheti
Dec 12 '18 at 16:36
add a comment |
Did you pay attention at "reflex"? According to the question which I think I've solved correctly, $ABCD$ is a quadrilateral whose vertex $D$ can be thought of as lying on the center of a circle with the other three lying on the circumference.
– Sameer Baheti
Dec 12 '18 at 16:36
Did you pay attention at "reflex"? According to the question which I think I've solved correctly, $ABCD$ is a quadrilateral whose vertex $D$ can be thought of as lying on the center of a circle with the other three lying on the circumference.
– Sameer Baheti
Dec 12 '18 at 16:36
Did you pay attention at "reflex"? According to the question which I think I've solved correctly, $ABCD$ is a quadrilateral whose vertex $D$ can be thought of as lying on the center of a circle with the other three lying on the circumference.
– Sameer Baheti
Dec 12 '18 at 16:36
add a comment |
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Is this a circular quadrilateral (has a circumcircle?)
– user376343
Dec 12 '18 at 16:31