Question

4
m∠bca =

m∠abc =

grade 8 • lesson 7 page 1 of

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Explanation:

Step1: Identify angle relationships

Since \( AD \) is a vertical line and \( AB \) is horizontal (implied by the right angle - like structure), \( \angle DAB = 90^\circ\) adjacent to the \( 97^\circ\) angle? Wait, no, actually, \( \angle DAA?\) Wait, looking at the diagram, \( \angle DAB \) and the \( 97^\circ\) angle: Wait, \( \angle DAA \) is not, but \( \angle DAB \) is a straight line? Wait, no, \( AD \) is vertical, \( AB \) is horizontal, so \( \angle DAB = 90^\circ\)? Wait, no, the angle between \( AD \) (vertical) and \( AC \) is \( 97^\circ\), and \( AB \) is horizontal, so \( \angle BAC \) is supplementary to \( 97^\circ\)? Wait, no, \( \angle DAB \) is a right angle? Wait, maybe \( AB \) and \( AD \) are perpendicular, so \( \angle DAB = 90^\circ\), then the angle between \( AC \) and \( AD \) is \( 97^\circ\), so the angle between \( AC \) and \( AB \) (i.e., \( \angle BAC \)) is \( 180^\circ - 97^\circ= 83^\circ\)? Wait, no, in triangle \( ABC \), the sum of angles is \( 180^\circ\). Also, \( \angle ABC=(14x - 1)^\circ\), \( \angle BCA=(2x + 2)^\circ\), and \( \angle BAC = 83^\circ\) (since \( 97^\circ\) and \( \angle BAC \) are supplementary as they form a linear pair? Wait, \( AD \) is a straight line, so \( \angle DAA?\) Wait, the angle at \( A \): \( \angle DAC = 97^\circ\), and \( \angle BAC \) is adjacent, so \( \angle BAC=180^\circ - 97^\circ = 83^\circ\). Then in triangle \( ABC \), \( \angle BAC+\angle ABC+\angle BCA = 180^\circ\). So \( 83+(14x - 1)+(2x + 2)=180\).

Step2: Solve for \( x \)

Simplify the equation: \( 83 + 14x-1 + 2x + 2=180\)
Combine like terms: \( (14x + 2x)+(83 - 1 + 2)=180\)
\( 16x+84 = 180\)
Subtract 84 from both sides: \( 16x=180 - 84 = 96\)
Divide by 16: \( x=\frac{96}{16}=6\)

Step3: Find \( m\angle BCA \)

Substitute \( x = 6\) into \( (2x + 2)^\circ\): \( 2(6)+2=12 + 2 = 14^\circ\)

Step4: Find \( m\angle ABC \)

Substitute \( x = 6\) into \( (14x - 1)^\circ\): \( 14(6)-1=84 - 1 = 83^\circ\)

Answer:

\( m\angle BCA = 14^\circ\)
\( m\angle ABC = 83^\circ\)