Unit 1, Lesson 7
Evidence, Angles, and Proof
Integrated Math 2
7.1
Warm-up
Mentally evaluate all of the missing angle measures in each figure.
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Mentally evaluate all of the missing angle measures in each figure.
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.”
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.
In many situations, it is important to understand the reasons why an idea is true. Here are some questions to ask when trying to convince ourselves or others that a statement is true:
In this lesson, we reasoned that pairs of vertical angles are always congruent to each other:
We saw this by labeling the diagram and making precise arguments having to do with transformations or angle relationships. For example, label the diagram with points:
Rotate the figure 180 degrees around point . Then ray goes to ray , and ray goes to ray . That means the rotation takes angle onto angle , so angle is congruent to angle .
Many true statements have multiple explanations. Another line of reasoning uses angle relationships. Notice that angles and together form line . That means that . Similarly, . That means that both and are equal to , so they are equal to each other. Since angle and angle have the same degree measure, they must be congruent.