Fishtank Learning

8th Grade Math
Unit 3: Transformations and Angle Relationships
Lesson 21

Objective

Define and use the exterior angle theorem for triangles.

Warm Up

Are the two triangles the same? Explain how you know.

Two triangles with sides labeled 3, 4, and 30° angles.

Anchor Problems - Problem 1

a. In each triangle, find the measure of the angle marked by $$x$$, $$y$$, or $$z$$.

Triangle with angles 33° and 48°; another with angles 31° and 14°; third with angles 40° and 70°; find angles x, y, and z.

b. In general, how can you find the measure of the triangle exterior angle $$c$$?

$Triangle with exterior angle labeled as $c^\circ$.$

Purpose

This problem introduces students to the exterior angle theorem of triangles. Using angle relationships they have learned so far, they find an exterior angle in three different triangles, and seek a general relationship between the exterior angle and two interior angles of any triangle.

Facilitation Guidance

Launch the problem by having students make 1-2 observations about each of the four triangles. Students can do a quick share with a peer sitting next to them.
Give students time to explore and solve Part (a) working in groups of 2 or 3. In each triangle, students are given two interior angles that are opposite the exterior angle in question. By solving for this exterior angle repeatedly in differently presented triangles, students combine a common structure and repeated reasoning that will support them in making a generality in Part (b). MP.7 Look for and make use of structure. MP.8 Look for and express regularity in repeated reasoning. Ask guiding questions as you circulate:
- What angle relationships are you using that are helping you find the angle measurements?
- What is the significance of $$180^{\circ}$$ in your work?
- Do you notice any repeated reasoning as you work through the three triangles?
- How can you organize your work to see any patterns or draw any conclusions?
Depending on how students do with Part (a), you may opt to debrief Part (a) before students work on Part (b), or you may have students shift directly into Part (b) and debrief at the end. If students struggled to see the similar nature in the three triangles and instead approached each triangle as if it were a different situation, then you may want to engage in a quick discussion and highlight the similarities and repetition in solution patterns. Then give students time to work on Part (b).
Debrief the problem by inviting students to share their general conclusions. Students may have a variety of explanations here, ranging from concrete to abstract.
- If students offer a concrete explanation, such as using numbers for the variables, validate these cases as additional examples that fit with Part (a) of the problem, however, here, you’re looking for a generality that would work for any triangle.
- One possible algebraic generalization is shown in the Student Response.
- Another algebraic generalization is to name the third angle of the triangle $$d$$. The interior angle sum theorem says: $$a+b+d=180$$. By supplementary angle relationships, you also have: $$c+d=180$$. Then by the transitive property, $$a+b+d=c+d$$, which implies $$a+b=c$$.
- Define this as the exterior angle theorem, and define an exterior angle as an angle created by one side of the polygon and a line extended from an adjacent side.

Anchor Problems - Problem 2

Triangle $${ABC}$$ has an interior angle that measures $${45^{\circ}}$$ and an exterior angle that measures $${129^{\circ}}$$.
What is the measure of the interior angle marked by $$x$$?

Triangle ABC with angles 45° and 129°,求x度数。

Purpose

The purpose of this problem is to give students the opportunity to put the exterior angle theorem into practice. Students may continue to use other angle relationships, such as supplementary angles and interior angle sum of a triangle, to find the value of $$x$$, but they also see how the exterior angle theorem is a more efficient approach.

Facilitation Guidance

Launch the problem by reviewing some vocabulary as they relate to the diagram. Have students point to an example of each description that you read aloud. If there is no example in the diagram, then they do not point to anything. Students can quietly look at a neighbor’s desk to check what they are doing and to have a neighbor check their understanding.
- Two different interior angles
- Two different exterior angles
- A pair of corresponding angles
- A pair of supplementary angles
- Two angles that are congruent
Give students time to work with a partner. At this point in the unit, there are two main approaches students may take if they are using angle relationships (although there may be others as well).
- Students may opt to use the internal angle sum theorem to add up all three angles inside the triangle and set it equal to 180. To do this, they first use a supplementary relationship between the angle marked $$129^{\circ}$$ and the adjacent interior angle. While this is a valid approach, it requires a few steps to find the value of $$x$$.
- Students may apply the exterior angle theorem they learned in Anchor Problem 1. They identify $$129^{\circ}$$ as the exterior angle which is equal to the sum of x and the $$45^{\circ}$$ angle. This is a more efficient approach as it only requires one step to find $$x$$.
- If you see both approaches being used as students work, take note of which pairs use which one to discuss in the debrief.
Debrief the problem by inviting students to share their thinking and response. As noted above, if different approaches were taken, showcase both for discussion. Note that both approaches can be used to find the value of x, challenging the right/wrong binary or the belief that there is only one way to solve a problem CRMT Culturally Responsive Math Teaching . At the same time, it is helpful to discuss how not all approaches are as efficient as others.

Problem Set

View Problem Set

Target Task

Triangle $${ABC}$$ is shown in the diagram below. Show two different ways that you can determine the measure of $$\angle {ABC}$$.

Triangle ABC with angles labeled and given measures: ∠BAC = 131° and ∠BCF = 120°.