Solving Angle Relationship Problems Lesson Plan

Warm Up

Give students time to answer these questions to help review the angle relationships they already know about. Allow students to check their work with a partner or table group.

The most common error students will make is subtracting from `180` for complementary angles or subtracting from `90` for supplementary angles. Students may also need a reminder that something like “`m∠2`” is read as “measure of angle `2`”.

Introducing linear pairs and adjacent angles

With this example, the goal is to let students’ prior knowledge recognize that the two angles are supplementary to help them find the measure of the angle. 

To help connect this information to linear pairs and adjacent angles, you can let students know that some angle pairs have multiple names that describe them. 

• Adjacent angles: Ask students what they notice about the two angles that are formed besides their measures. Students should recognize that they share ray `BC`. When students recognize that the angles are next to each other and share a ray, you can let them know that we call them “adjacent angles” because they are next to each other, like neighbors.
• Linear pair: Remind students that these two angles form a line. Let students know that linear pairs are different from supplementary angles because the angles must be adjacent angles.

Introducing vertical angles

Allow students a moment to list any angle pairs they see. The goal is to help students practice identifying linear pairs and adjacent angles, like `∠1` and `∠2`.

Once students have identified the different combinations, let students know that there is one more angle relationship: vertical angles. To have students use their critical thinking skills, consider telling them that `∠1` and `∠3` form vertical angles, as well as `∠2` and `∠4`. Allow students to describe it in their own words; however, they should recognize that the angles are opposite of each other when there are two lines that form an `\text{X}`.

Discovering the relationship with vertical angles

Give a bunch of problems with two intersecting lines and one angle measure given. Ask students if they can use their knowledge of linear pairs to figure out the measure of all the other angles.

Students will reason that `∠1` and `∠2` are linear pairs, so `m∠2` must be `138^\circ`. Similarly `m∠4` must be `138^\circ`. They will also reason that `∠2` and `∠3` are linear pairs, so `m∠2` must be `42^\circ`. Some students might say that `∠1` and `∠3` must be congruent because they look the same. You have to remind students that in geometry, we do not go by how things look but only by what is given.

Once students have done a few such problems, they will recognize that vertical angles are always equal in measure. Depending on time and your inclination, you can also push their thinking by doing some algebra. 

Students might reason that `m∠2` and `m∠4` must be `(180-x)^\circ`. And that `m∠3` must be `180 - (180-x)^\circ` which would be `x^\circ`. 

Straight angles and Full angles

While these angles are quite straightforward, it is best to formally introduce them so that students can start using this language in their reasoning.

It may be helpful to relate straight angles and full angles to examples they have already seen with linear pairs and vertical angles. This can help students recognize that they have already been using similar information in a different way to solve problems.

Solving angle relationship problems

Allow students time to work on the problems independently before checking with a partner or table group to try and solve each problem. Encourage students to use everything they know at this point about angles and their relationships to find the value of the variables.

Students should be able to solve each problem, but students may still have doubts for each:

• First example: Students should be familiar; however, using three variables may cause some students to think it is more complicated than it actually is. 
• Second example: Students should ideally recognize the right angle symbol. Students will likely find `m` by calculating `180 - 90 - 23`; however, some students will recognize that `m` is complementary with `23^\circ` and only calculate using `90-23`. 
• Third example: Students may be tempted to say that `n` is a vertical angle to `80^\circ`. In this case, remind students that vertical angles are created when two lines intersect; however, this image is created using `4` rays. Encourage students to think about what the sum of all the angles combined would be. Students should recognize it would be `360^\circ`, or a full angle.

Solving angle relationship problems with algebraic expressions

With this example, students may not recognize the congruent symbols for the angles. They should recognize that the angles have a sum of `180^\circ`. Once students understand that the third angle is also `k`, give them an opportunity to try the problem.

Different solving methods

Students may approach this problem using different methods, so encourage students to explain how they found `k`. For example, some students may find that `180-30 = 150`. From there, since there are `2` angles that have the same measure, they will divide `150` by `2`. Other students may recognize that they could write and solve the equation `2k + 30 = 150`, or other equivalent variations. Encourage students to practice writing equations when solving angle relationship problems to help deepen their understanding.