∠ABD and ∠DBC form a linear pair, and ∠ABD and ∠CBE are vertical angles. If m∠ABD=2(5x−11), m∠DBC=2x+10, and m∠CBE=23y+36, select all the statements that are truex=16 y=153 m∠CBE=42 m∠ABD=138 m∠ABE=138 ∠DBC and ∠ABE are vertical angles.
Accepted Solution
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Answer:True: x=16[tex]m\angle ABD=138^{\circ}[/tex]∠DBC and ∠ABE are vertical angles.Step-by-step explanation:A linear pair of angles is formed when two lines intersect. Two angles are said to be linear angles if they are adjacent angles formed by two intersecting lines.Angles ∠ABD and ∠DBC form a linear pair, then they are supplementary and [tex]m\angle ABD+m\angle DBC=180^{\circ}[/tex]Angles ∠ABD and ∠CBE are vertical angles, then they are congruent and[tex]m\angle ABD=m\angle CBE[/tex] If [tex]m\angle ABD=2(5x-11), \ m\angle DBC=2x+10,[/tex] and [tex]m\angle CBE=23y+36,[/tex] then[tex]m\angle ABD+m\angle DBC=180^{\circ}\Rightarrow 2(5x-11)+2x+10=180\\ \\10x-22+2x+10=180\\ \\12x=192\\ \\x=16\\ \\m\angle ABD=2(5\cdot 16-11)=138^{\circ}\\ \\m\angle DBC=2\cdot 16+10=42^{\circ}[/tex]Now,[tex]m\angle ABD=m\angle CBE[/tex][tex]138^{\circ}=m\angle CBE\Rightarrow 23y+36=138\\ \\23y=102\\ \\y=\dfrac{102}{23}[/tex]True: x=16[tex]m\angle ABD=138^{\circ}[/tex]∠DBC and ∠ABE are vertical angles.