We are given that ∠ ABC is a right angle. Therefore, it has a measure of 90.
Statement
∠ ABC is a right angle, so m ∠ ABC = 90.
We can tell that the ray BR divides ∠ ABC into ∠ 1 and ∠ 2. Thus, by the Angle Addition Postulate m ∠ 1 + m ∠ 2 = m ∠ ABC.
Statement
By the Angle Addition Postulate, m ∠ 1 + m ∠ 2 = m ∠ ABC.
We know that m ∠ ABC= 90, and we are given that m ∠ 1 = 45. Notice that we can use the Substitution Property of Equality and substitute these values into our equation. Then, we get 45+ m ∠ 2= 90.
Statement
By Substitution Prop. of Equality, 45+ m ∠ 2= 90.
Now, we can use the Subtraction Property of Equality and subtract 45 from both sides of the equation. This gives us m ∠ 2 = 45.
Statement
By Subtraction Prop. of Equality, m ∠ 2= 45.
Notice that the measures of ∠ 1 and ∠ 2 are the same. Therefore, they are congruent by the definition of congruent angles.
Statement
∠ 1 and ∠ 2 are congruent by the definition of congruent angles.
We can tell that BR divides ∠ ABC into two congruent angles. Thus, by the definition of an angle bisector, BR bisects ∠ ABC, which is what we wanted to prove!
Statement
By the definition of an angle bisector, BR bisects ∠ ABC.
Completed Proof
∠ ABC is a right angle, so m ∠ ABC=90. By the Angle Addition Postulate, m ∠ 1 + m ∠ 2 = m ∠ ABC. By the Substitution Property of Equality, 45+ m ∠ 2 = 90. By the Subtraction Property of Equality, m ∠ 2 = 45. ∠ 1 and ∠ 2 are congruent by the definition of congruent angles. By the definition of an angle bisector, BR bisects ∠ ABC.
