Elena and Diego are working together on this problem: Here is a figure where ray meets line . The dashed rays are angle bisectors.

Diego makes this conjecture: “The angle formed between the angle bisectors is always a right angle, no matter what the angle between and is.”

Elena says, “It’s difficult to tell specifically which angles you’re talking about.” She labels the diagram and restates the conjecture as the following: “Ray bisects angle into 2 congruent angles. Ray bisects angle into 2 congruent angles. We conjecture angle is a right angle.”

Diego adds more information to the diagram as he tells Elena, “We can put letters here to represent the angle measures. So these 2 angles are each , and these are . That means our conjecture is .”

Elena exclaims, “Oh! I see it now. Angle is 180 degrees, so . Then the middle part has to be a right angle.”

Diego writes down a summary of their conversation: “For any straight line and ray , the angle bisectors of the 2 angles form a right angle. That’s because there are 2 pairs of congruent angles,  and , that sum to 180. So has to equal 90 degrees, a right angle.”

1. Take turns with your partner to explain what is happening in this conversation.
   1. For each paragraph that you read, explain to your partner what Elena or Diego is saying and doing.
   2. For each paragraph that your partner reads, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.
2. Diego says, “These 2 angles are each .” Which 2 angles is he referring to? How does he know they are the same size?
3. Why does have to equal 90 degrees?

Here are 2 intersecting lines that create 2 pairs of vertical angles:

1. What is the relationship between vertical angles? Write down a conjecture. Label the diagram to make it easier to write your conjecture precisely.

2. How do you know your conjecture is true for all possible pairs of vertical angles? Explain your reasoning.