"If ray BC lies within angle ABD, then angle ABC is congruent to angle DBC" sometimes, never, or always !!??

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24-09-2021
Mathematics

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"If ray BC lies within angle ABD, then angle ABC is congruent to angle DBC" sometimes, never, or always !!??

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akposevictor akposevictor · Answer 1 · 2021-09-29T19:02:27+02:00

[tex]\angle ABC[/tex] can be congruent to [tex]\angle DBC[/tex] sometimes, if ray BC is an angle bisector of [tex]\angle ABC[/tex].

In the diagram attached below shows two possible figures we can assume to get if ray BC is lies within [tex]\angle ABD[/tex].

We are assuming that [tex]\angle ABD = 50^{\circ}[/tex]

In figure 1, if ray BC lies within [tex]\angle ABD[/tex], we can have two angles that are not congruent to each other. We would have:

[tex]m \angle DBC = 30^{\circ}\\\\m \angle ABC = 20^{\circ}[/tex]

This implies that [tex]\angle ABC[/tex] will not always be congruent to [tex]\angle DBC[/tex] if ray BC lies within [tex]\angle ABD[/tex].

In figure 1, if ray BC is an angle bisector and it lies within [tex]\angle ABD[/tex], we can have two angles that are congruent to each other. We would have:

[tex]m \angle DBC = 25^{\circ}\\\\m \angle ABC = 25^{\circ}[/tex]

This implies that [tex]\angle ABC[/tex] can sometimes be congruent to [tex]\angle DBC[/tex] if ray BC is an angle bisector that lies within [tex]\angle ABD[/tex].

Therefore, we can conclude that:

[tex]\angle ABC[/tex] can be congruent to [tex]\angle DBC[/tex] sometimes, if ray BC is an angle bisector of [tex]\angle ABC[/tex].

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