The four angles of a quadrilateral form an arithmetic sequence. The largest is 15 degrees less than twice the smallest. What is the degree measure of the largest angle?
+26367 The four angles of a quadrilateral form an arithmetic sequence.
The largest is 15 degrees less than twice the smallest.
What is the degree measure of the largest angle?
The four angles of a quadrilateral are \(\alpha_1,\ \alpha_2,\ \alpha_3,\ \alpha_4\)
The smallest angle is \(\alpha_1\)
The largest angle is \(\alpha_4\)
The common distance of the arithmetic sequence is d
\(\begin{array}{|rcll|} \hline \mathbf{\text{arithmetic sequence}} \\ \hline \alpha_2 &=& \alpha_1 +d \\ \alpha_3 &=& \alpha_1 +2d \\ \alpha_4 &=& \alpha_1 +3d \\\\ \mathbf{\alpha_1+\alpha_2+ \alpha_3+ \alpha_4} &=& \mathbf{360^\circ} \\ \alpha_1+(\alpha_1 +d)+ (\alpha_1 +2d)+ (\alpha_1 +3d) &=& 360^\circ \\ 4\alpha_1+6d &=& 360^\circ \quad | \quad : 2 \\ 2\alpha_1+3d &=& 180^\circ \\ \alpha_1+\alpha_1+3d &=& 180^\circ \quad | \quad \mathbf{\alpha_4 = \alpha_1 +3d} \\ \alpha_1+\alpha_4 &=& 180^\circ \quad | \quad \mathbf{\alpha_4 = 2\alpha_1 -15^\circ } \\ \alpha_1+2\alpha_1 -15^\circ &=& 180^\circ \\ 3\alpha_1 -15^\circ &=& 180^\circ \quad | \quad : 3 \\ \alpha_1 -5^\circ &=& 60^\circ \\ \alpha_1 &=& 60^\circ+5^\circ \\ \mathbf{\alpha_1} &=& \mathbf{65^\circ} \\ \hline \mathbf{\alpha_4} &=& \mathbf{2\alpha_1 -15^\circ} \\ \alpha_4 &=& 2\cdot 65^\circ -15^\circ \\ \alpha_4 &=& 2\cdot 65^\circ -15^\circ \\ \alpha_4 &=& 130^\circ -15^\circ \\ \mathbf{\alpha_4} &=& \mathbf{115^\circ} \\ \hline \end{array}\)
The largest angle is \(\mathbf{115^\circ}\)
