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If in triangle \(ABC\) point \(D\) lies on side \(AC\), what is the degree measure of \(\angle BCA\) ?
(1) The degree measure of \(\angle ABD\) is half that of \(\angle BDC\).
(2) The degree measure of \(\angle DBC\) is 10°.
(1) The degree measure of \(\angle ABD\) is half that of \(\angle BDC\).
(2) The degree measure of \(\angle DBC\) is 10°.
16 Sep 2014, 01:10
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Official Solution:
If in triangle \(ABC\) point \(D\) lies on side \(AC\), what is the degree measure of \(\angle BCA\) ?
(1) The degree measure of \(\angle ABD\) is half that of \(\angle BDC\).
Let \(\angle ABD = x\) degrees. Then, \(\angle BDC = 2x\) degrees. However, this statement doesn't tell us anything about \(\angle BCA\) directly. Not sufficient.
(2) The degree measure of \(\angle DBC\) is 10°.
But again, this doesn't give us direct information about \(\angle BCA\) directly. Not sufficient.
(1)+(2) Given that \(\angle BDC\) is an exterior angle to triangle \(ABD\), it follows that \(\angle BDC = \angle BAC + \angle ABD\) (the measure of an exterior angle of a triangle is equal to the sum of two remote interior angles.). So, we have that \(2x = \angle BAC + x\), from which we it follows that \(\angle BAC = x\). Next, since the angles in a triangle sum to 180°, we get: \(\angle BCA + \angle ABC + \angle BAC = \angle BCA + (x + 10) + x = 180°\). Without knowing the value of \(x\), we cannot determine \(\angle BCA\). Not sufficient.
Answer: E
If in triangle \(ABC\) point \(D\) lies on side \(AC\), what is the degree measure of \(\angle BCA\) ?
(1) The degree measure of \(\angle ABD\) is half that of \(\angle BDC\).
Let \(\angle ABD = x\) degrees. Then, \(\angle BDC = 2x\) degrees. However, this statement doesn't tell us anything about \(\angle BCA\) directly. Not sufficient.
(2) The degree measure of \(\angle DBC\) is 10°.
But again, this doesn't give us direct information about \(\angle BCA\) directly. Not sufficient.
(1)+(2) Given that \(\angle BDC\) is an exterior angle to triangle \(ABD\), it follows that \(\angle BDC = \angle BAC + \angle ABD\) (the measure of an exterior angle of a triangle is equal to the sum of two remote interior angles.). So, we have that \(2x = \angle BAC + x\), from which we it follows that \(\angle BAC = x\). Next, since the angles in a triangle sum to 180°, we get: \(\angle BCA + \angle ABC + \angle BAC = \angle BCA + (x + 10) + x = 180°\). Without knowing the value of \(x\), we cannot determine \(\angle BCA\). Not sufficient.
Answer: E
