Question

complete the indirect proof of the exterior angle inequality theorem. 5.3–5.4
given: the figure
prove: ( mangle 1 > mangle y )

1. step 1 assume ____________.
2. step 2 case 1: ( mangle 1 = mangle y )

since ____________ (exterior angle theorem),
__________ (substitution), and __________.

1. case 2: ( mangle 1 < mangle y )

since ____________ (exterior angle theorem),
( mangle z = ) __________ (addition property of equality), and __________

1. step 3 but ( mangle z > 0 ); ( \therefore ) ____________.

Explanation:

Step 1: State indirect proof assumption

Assume the opposite of the conclusion: $m\angle 1 \leq m\angle Y$.

Step 2: Case 1 ($m\angle1=m\angle Y$)

By Exterior Angle Theorem: $m\angle1=m\angle Y+m\angle Z$. Substitute $m\angle1=m\angle Y$: $m\angle Y=m\angle Y+m\angle Z$. Simplify: $m\angle Z=0$, which is impossible (angle measure >0).

Step 3: Case 2 ($m\angle1

By Exterior Angle Theorem: $m\angle1=m\angle Y+m\angle Z$. Rearrange: $m\angle Z=m\angle1 - m\angle Y$. Since $m\angle1

Step 4: Conclude contradiction

Both cases lead to contradictions, so the assumption is false.

Answer:

1. $m\angle 1 \leq m\angle Y$
2. $m\angle1=m\angle Y+m\angle Z$; $m\angle Y=m\angle Y+m\angle Z$; $m\angle Z=0$ (impossible)
3. $m\angle1=m\angle Y+m\angle Z$; $m\angle1 - m\angle Y$; $m\angle Z<0$ (impossible)
4. The assumption is false, so $m\angle1>m\angle Y$