Lesson 4

In order to make sense of trigonometry, students must be certain that right triangles with the same angle measures are similar, and therefore that ratios of side lengths in right triangles with the same angle measure, regardless of the size of the triangle, are equal. This warm-up challenges students to verify that ratios of side lengths in right triangles with the same angles are equal. Students are given the measure of only one of the acute angles, so they must rely on the Angle-Angle Triangle Similarity Theorem.

Ask students to convince someone who thought \(\frac{b}{d}\) was greater that in fact the two ratios are equal. Invite several students to share different ways of arguing that the fractions are equal, such as:

• measure the lengths of \(a, b, c, \text{ and } d\) and compare the fractions
• the triangles are similar by the Angle-Angle Triangle Similarity Theorem, so \(\frac{a}{b} = \frac{c}{d}\) and that means that \(\frac{a}{c} = \frac{b}{d}\) too
• the triangles are similar by the Angle-Angle Triangle Similarity Theorem, so \(b=ka \text{ and } d=kc\) for some scale factor \(k\), so \(\frac{a}{c} = \frac{ka}{kc} = \frac{b}{d}\).
• the triangles are similar by the Angle-Angle Triangle Similarity Theorem, so side lengths are proportional within and between triangles, so \(\frac{a}{c} = \frac{b}{d}\)

In this activity, students begin to build their own table of trigonometric ratios (ratios of side lengths in right triangles, by angle measure of one acute angle). For now, call this the right triangle table so students remember these ratios only apply to right triangles. Building these ratios from the similar triangles they are based on helps students build connections between similar triangles and trigonometry.

Measuring several examples of triangles with the same angle measurements is essential in both the digital and paper versions of this activity. In the paper version, it provides more data points to get a more accurate ratio for the given angle. In both versions, it helps students understand that all right triangles with their angle as one of the angles will be similar and thus have the same ratios.

In the next activity and the cool-down of this lesson, students will look for patterns in the right triangle table, and also apply the ratios in the right triangle table to find unknown angle measures given a ratio of side lengths. Do not use the word trigonometry or any of the names of the ratios yet.

Some students might struggle to identify opposite and adjacent legs. Prompt students to highlight the angle in question to accurately identify sides. If you point to “your angle” then the leg you are touching is adjacent and the leg you are not touching is opposite. The hypotenuse is always the longest side.

Invite groups to share what they noticed and wondered about their data. (Each ratio is about the same for the angle. The ratio of the opposite side to the hypotenuse for angle \(A\) is the same as the ratio of the adjacent side to the hypotenuse for angle \(B\).)

In the previous activity, students noticed that if right triangles have the same angle measures, they have the same ratios among their side lengths. Students may also have noticed patterns between the columns, such as that when two angles are complementary, there are relationships between the ratios associated with them.

In this activity, students look for patterns in the right triangle table, including patterns that let them make predictions about other data points. Monitor for students who:

• calculate or estimate the ratio for a 35 degree angle by halving a 70 degree angle
• calculate or estimate the ratio for a 35 degree angle using the 30 and 40 degree angles
• use complementary relationships and find the measures of a 35 degree angle using the 55 degree angle
• draw and measure a right triangle with a 35 degree angle

Asking students what they know about the angle in a triangle with a certain ratio comes up again in the cool-down.

Display the class table and distribute the blackline master with the completed table. Students will need their completed right triangle table for the next several lessons. Suggest students tape it into their workbook or staple it to their reference chart. They will only need to reference it during this unit.

Ask students, "What do you notice? What do you wonder?" Record and display their responses for all to see. If possible, record the relevant reasoning on or near the table. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. 

Things students may notice:

• The angles are increasing by multiples of ten.
• The values in the opposite \(\div\) hypotenuse column are increasing.
• The values in the first two ratio columns are all less than one.
• The first two ratio columns are the same numbers but backwards.

Things students may wonder:

• Do all complementary angles have the same ratios?
• What happens if the angle is in between these?
• What happens if the angle is larger than 90?

If students are struggling to make the connection between the 35 and 55 degree angles prompt them to draw a right triangle with a 35 degree angle.

The goal of this activity is to continue to introduce students to patterns in the right triangle table. Students need not resolve or master concepts like the relationships between complementary angles at this point.

Invite a student who calculated the 35 degree angle ratios by halving the 70 degree angle ratios to share. Tell students, “That's a great conjecture! It seems like if an angle is half the size of another, we could divide the values of the ratios associated with the bigger angle in half to find the values of the ratios associated with the smaller angle. Let’s test that hypothesis using other angles on the chart.” (The conjecture is false, the ratios associated with the 40 degree angle do not have half the value of the ratios associated with the 80 degree angle.)

Invite a student who used the 30 and 40 degree angles to estimate or calculate the 35 degree angles to share their method as a conjecture. “Let’s test that hypothesis using other angles on the chart.” (Yes, at least on this table, if one angle measure is between two other angle measures, then the values of the associated ratios will be between the values of the ratios associated with the larger and smaller angles. No, the values of the ratios associated with a 50 degree angle are not exactly the midpoint of the values of the ratios associated with a 40 and a 60 degree angle.)

Invite a student who used the complementary relationship between 55 degrees and 35 degrees to reason about the ratios of the adjacent side to the hypotenuse or the opposite side to the hypotenuse to share their method as a conjecture. “Let’s test that hypothesis using other angles on the chart.” (\(20 + 70 = 90\), and for the 20 degree and 70 degree angles, the values of the ratios in the first two columns are swapped, so it seems like the same pattern would work here.) Remind students that we define complementary angles as two angles whose measures add up to 90 degrees.

Invite a student who made a right triangle with a 35 degree angle to confirm which solution methods matched their measurements.