Question

∠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.

Answer 1

Answer:

True:

x=16

$m\angle ABD=138^{\circ}$

∠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

$m\angle ABD+m\angle DBC=180^{\circ}$

Angles ∠ABD and ∠CBE are vertical angles, then they are congruent and

$m\angle ABD=m\angle CBE$

If $m\angle ABD=2(5x-11), \ m\angle DBC=2x+10,$ and $m\angle CBE=23y+36,$ then

$m\angle ABD+m\angle DBC=180^{\circ}\Rightarrow 2(5x-11)+2x+10=180\\ \\10x-22+2x+10=180\\ \\12x=192\\ \\x=16\\ \\m\angle ABD=2(5\cdot 16-11)=138^{\circ}\\ \\m\angle DBC=2\cdot 16+10=42^{\circ}$

Now,

$m\angle ABD=m\angle CBE$

$138^{\circ}=m\angle CBE\Rightarrow 23y+36=138\\ \\23y=102\\ \\y=\dfrac{102}{23}$

True:

x=16

$m\angle ABD=138^{\circ}$

∠DBC and ∠ABE are vertical angles.