Previously, students found numerous angle sizes by reasoning, without using a protractor. They have done so with problems with and without context. In this activity, students consolidate various skills and understandings gained in the unit and apply them to solve problems that are more abstract and complex. They rely, in particular, on their knowledge of right angles and straight angles to reason about unknown measurements. (Students may need a reminder that an angle marked with a small square is a right angle.)

The angles with unknown measurements are shaded but not labeled, motivating students to consider representing them (or their values) with symbols or letters for easier reference. Students also may choose to write equations to show how they are thinking about the problems.

When students use the fact that angles making a line add up to and that angles making a right angle add up to , they make use of structure to find the unknown angle measures (MP7).

• Display the angles. Select students to share their responses. Record and display their reasoning.
• Highlight equations that illuminate the relationship between the known angle, the unknown angle, and the reference angle (, , or ). For instance: , , , and so on.
• Label the diagrams with letters or symbols, as needed, to facilitate equation writing.
• When discussing the last question, highlight that finding unknown values sometimes involves multiple steps, and some steps may need to happen before others.

This activity features an Information Gap (Info Gap) routine in which students solve abstract multi-step problems involving an arrangement of angles with several unknown measurements. By now students have the knowledge and skills to find each unknown value, but the complexity of the diagram and the Info Gap structure demand that they carefully make sense of the visual information and look for entry points for solving the problems. Students need to determine what information is necessary, ask for it, and persevere if their initial requests do not yield the information they need (MP1). The process also prompts them to refine the language they use to ask increasingly more precise questions until they get useful input (MP6).

• Select students to share how they found each angle measure. Record their reasoning, and highlight equations that clearly show the relationships between angles.
• “Which angle measurements were easy to find? What made them easy?” (P and D, because it was fairly easy to see that each of them and a neighboring angle make a straight angle.)
• “Which ones were a bit more involved? Why?” (E, because there are 5 angles that meet at that point. We needed to find A or D before finding E.)