BC = 1/2 AC
2m angle DBC = m angle ABC
Angle ABC is bisected by ray BD
m∠WUT = 38°
Step-by-step explanation:
For both questions, we use the Angle Bisector Theorem.
Question 1:
We know since ray DB bisects ABC, AB and BC are congruent. They are split into 2 equal parts. Therefore, we know that if we take half of mAC, we should get either mBC or mAB. Therefore, the 1st statement is correct.
Since it is ray DB bisecting ABC, it created 2 right angles. If angle DBC was bisected, it would create 2 new angles that were 45° each, so the 2nd statement is incorrect.
We know that line ABC is equal to 180°. So if we add m∠ABD and m∠DBC (they both are 90°, as denoted in the picture) together, we should get 180°. Also, since they are congruent angles, one angle multiplied twice would also be 180°. Therefore, the 3rd statement is correct.
The Angle Bisector Theorem doesn't state that the bisector and one segment would be congruent. The length of the bisector could be anything and is not marked congruent to the other segments. Therefore, the 4th statement is incorrect.
Since we see that there is a bisector DB bisecting line ABC, creating 2 congruent segments and angles, we know that the 5th statement is correct.
Question 2:
The Angle Bisector Theorem states that a bisector that bisects 2 angles will become 2 smaller but congruent angles. So,
Step 1: Find x
4x + 6 = 6x - 10
6 = 2x - 10
16 = 2x
x = 8
Step 2: Find m∠WUT
m∠WUT = 6x - 10
m∠WUT = 6(8) - 10
m∠WUT = 48 - 10
m∠WUT = 38°
