Let's begin by drawing ∠ WXZ and constructing ∠ YXZ congruent to ∠ WXZ. Then we will make a conjecture as to the measure of ∠ WXY and prove it.
Drawing ∠ WXZ and Constructing ∠ YXZ
Let's draw ∠ WXZ such that m ∠ WXZ=45.
Now, we can construct ∠ YXZ that is congruent to ∠ WXZ. We will need XZ for this. Let's place the tip of the compass at point X and draw a large arc that intersects both sides of ∠ WXZ, and goes below XZ.
Next, we will place the point of our compass on the point of intersection, and adjust so that the pencil tip is on the other point of intersection.
Now, without changing the compass setting, we draw an arc below XZ that intersects the larger arc that we drew.
Finally, we can draw XY that passes through the point of intersection of the arcs.
Making and Proving the Conjecture
Since ∠ YXZ is congruent to ∠ WXZ, the ray XY is an angle bisector of ∠ WXY. The measure of both ∠ WXZ and ∠ YXZ is 45. Therefore, our conjecture will be that m ∠ WXY=90, which means that is it a right angle. To prove it, let's begin by looking at the given information and the desired outcome of the proof.
Given:& XZ bisects ∠ WXY, m ∠ WXZ =45.
Prove:& ∠ WXY is a right angle
We are given that XZ bisects ∠ WXY and that m ∠ WXZ=45. This is how we will begin our proof!
Statement 1)& XZ bisects ∠ WXY, & m ∠ WXZ =45
Reason 1) & Given
We know that an angle bisector divides an angle into two congruent angles, which means that ∠ WXZ ≅ ∠ YXZ.
Statement 2)& ∠ WXZ ≅ ∠ YXZ
Reason 2) & Definition of an angle bisector
By the definition of congruent angles we know that m ∠ WXZ= m ∠ YXZ.
Statement 3)& m ∠ WXZ = m ∠ YXZ
Reason 3) & Definition of congruent angles
We know that m ∠ WXZ =45. By the Substitution Property of Equality, we can substitute it in our equation. This gives us 45= m ∠ YXZ.
Statement 4)& 45= m ∠ YXZ
Reason 4) & Substitution Prop. of Equality
Notice that X is the interior of ∠ WXY. Therefore, by the Angle Addition Postulate m ∠ WXY = m ∠ WXZ + m ∠ YXZ.
Statement 5)& m ∠ WXY =m ∠ WXZ + m ∠ YXZ
Reason 5) & Angle Addition Postulate
We know that m ∠ WXZ =45 and that m ∠ YXZ= 45. Therefore, by the Substitution Property of Equality, we can substitute these values in our equation, m ∠ WXY=45+ 45.
Statement 6)& m ∠ WXY=45+45
Reason 6) & Substitution Prop. of Equality
Now, we can simplify the right-hand side by calculating the sum of 45 and 45. It is 90, and hence we get the measure of our angle, m ∠ WXY = 90.
Statement 7)& m ∠ WXY=90
Reason 7) & Simplify
By the definition of a right angle, we can tell that ∠ WXY is a right angle. This is what we wanted to prove!
Statement 8)& ∠ WXY is a right angle
Reason 8) & Definition of a right angle
