I have triangle XYZ with BX as the bisector sides XY and XZ are congruent. I have to write a proof for angle Y being congruent to angle Z.

This is what I have, but it doesn't seem correct - can anyone help?

XY is congruent to XZ - Given

XB bisects angle XYZ - Perpendicular Bisector of Isosceles Triangle

YB is congruent to BZ - Don't know why

XB is congruent to XB - Reflexive Prop

XBY is congruent to XBZ - SSS Theorem

Angle Y is congruent to Angle Z - CPCTC

You want to use ASA, not SSS

I would write it this way

XY is congruent to XZ - Given
angle YXB = angle ZXB - given angle x is bisected
XB = XB -reflexise prop.

then triangle XYB is congruent to triangle XZB (ASA)

therefore angle Y = angle Z (properties of congruent triangles)

line RS=line UT and line RT= line US

prove triangle RST= triangle UTS

Your proof is almost there, but there is one step that needs clarification. Let me help you go through the proof and explain each statement in detail:

Statement 1: XY is congruent to XZ - Given

This statement is given in the problem, so it doesn't require any explanation.

Statement 2: XB bisects angle XYZ - Perpendicular Bisector of Isosceles Triangle

When you state that XB bisects angle XYZ, you need to provide a reason to support this claim. The fact that the triangle is isosceles and XB bisects the angle at X allows you to conclude that XY and XZ are congruent.

Statement 3: YB is congruent to BZ - Don't know why

In order to show that YB is congruent to BZ, you can use the fact that XB is the bisector of angle XYZ. Since XB bisects the angle, it divides it into two congruent angles, say angle XBY and angle XBZ. Additionally, you know that XY is congruent to XZ from the given information. By using the SSS (Side-Side-Side) congruence theorem, you can conclude that triangle XBY is congruent to triangle XBZ. Since corresponding parts of congruent triangles are congruent, the lengths YB and BZ must also be congruent.

Statement 4: XB is congruent to XB - Reflexive Property

This statement is the reflexive property of equality, which states that any segment, in this case XB, is always congruent to itself.

Statement 5: XBY is congruent to XBZ - SSS Theorem

You correctly state that triangle XBY is congruent to triangle XBZ. The congruence is based on the lengths XY congruent to XZ (given), XB congruent to XB (reflexive property), and YB congruent to BZ (proved in statement 3).

Statement 6: Angle Y is congruent to Angle Z - CPCTC

CPCTC stands for "Corresponding Parts of Congruent Triangles are Congruent." Since triangle XBY is congruent to triangle XBZ, you can conclude that corresponding angles of these triangles are congruent. Thus, angle Y is congruent to angle Z.

So, your proof is nearly correct, but you should include a clear explanation for statement 3, which shows that YB is congruent to BZ based on the fact that XB bisects angle XYZ.