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.
