Question
The angle between the two altitudes of a parallelogram through the same vertex of an obtuse angle of the parallelogram is 30∘. The measure of the obtuse angle is
a) 100∘
b) 150∘
c) 105∘
d) 120∘
The angle between the two altitudes of a parallelogram through the same vertex of an obtuse angle of the parallelogram is 30∘. The measure of the obtuse angle is
a) 100∘
b) 150∘
c) 105∘
d) 120∘
Answer:
:
Let EC and FC be altitudes and ∠ECF=30∘.
Let ∠EDC=x=∠FBC
So, ∠ECD=90−x∘and∠BCF=90∘−x
So, by property of the parallelogram,
∠ADC+∠DCB=180∘∠ADC+(∠ECD+∠ECF+∠BCF)=180∘⇒x+90∘−x+30∘+90∘−x=180∘⇒−x=180∘−210∘=−30∘⇒x=30∘
Hence, ∠DCB=30∘+60∘+60∘=150∘.
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:
Let EC and FC be altitudes and ∠ECF=30∘.
Let ∠EDC=x=∠FBC
So, ∠ECD=90−x∘and∠BCF=90∘−x
So, by property of the parallelogram,
∠ADC+∠DCB=180∘∠ADC+(∠ECD+∠ECF+∠BCF)=180∘⇒x+90∘−x+30∘+90∘−x=180∘⇒−x=180∘−210∘=−30∘⇒x=30∘
Hence, ∠DCB=30∘+60∘+60∘=150∘.
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