Question

A conjecture and the two-column proof used to prove the conjecture are shown. Given: angle 1 is supplementary to angle 2. Ray B D bisects angle A B C. Prove: angle 1 is supplementary to angle 3. Ray B A and ray B C connected at point B form an angle labeled as A B C. A ray B D starts from point B, bisecting the angle A B C into two parts A B D and C B D. Angle A B D is labeled as 2 and angle C B D is labeled as 3. There is a separate angle labeled 1. Drag an expression or phrase to each box to complete the proof. Statement Reason 1. ∠1 is supplementary to ∠2. Given 2. Response area Definition of supplementary 3. BD−→− bisects ∠ABC. Given 4. m∠2=m∠3 Response area 5. Response area Response area 6. ∠1 is supplementary to ∠3. Definition of supplementary

Answer 1

The two-column proof shows that angle 1 is supplementary to angle 3 using given information, definitions of supplementary angles, and bisected angles to substantiate the reasoning through logical steps.

Step-by-step explanation:

To complete the two-column proof for the conjecture that angle 1 is supplementary to angle 3, we need to fill in the missing statements and reasons. Using the properties of bisected angles and the definition of supplementary angles, we can construct the proof as follows:

∠1 is supplementary to ∠2. - Given

m∠2 + m∠1 = 180° - Definition of Supplementary

BD bisects ∠ABC. - Given

m∠2 = m∠3 - Definition of Angle Bisector

m∠3 + m∠1 = 180° - Substitution (from step 2 and 4)

∠1 is supplementary to ∠3. - Definition of Supplementary (conclusion from step 5)

Each step builds upon the previous information to reach our conclusion that angle 1 is indeed supplementary to angle 3.