Unit 3, Lesson 15
Converse of the Pythagorean Theorem
Geometry
15.1
Warm-up
Standards Alignment
Building On
Addressing
HSG-SRT.B.4
Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
Building Toward
Instructional Routines
NoneMaterials
Activity Narrative
In this Warm-up, students play with two sides of a triangle with fixed length to recognize how the angle between them is connected to the opposite side of the triangle.
In the digital version of the activity, students use an applet to adjust an angle between two segments to see how the opposite side is affected. The applet allows students to more clearly see the side opposite the angle being adjusted and play with different options more dynamically. The digital version may be preferable if it is available and students would benefit from visualizing the connection.
Activity Synthesis
The purpose of the discussion is for students to understand that, when two side lengths of a triangle are fixed, the side opposite the angle between the two fixed sides is longer when the angle is large and shorter when the angle is small.
Invite students to share any connections they noticed between angle and the length of . If it does not come up, ask students what they notice happens to the length when the angle is increased.
Then, invite students to share anything else they notice about the length of segment . If it comes up, highlight any observations that mention the maximum or minimum possible lengths of . Students do not need to recognize the triangle inequality at this point.
Ask students,
- “What do we know about the length of when angle is a right angle?” ( by the Pythagorean Theorem.)
- “Is the door handle closer to the latch on the door frame when I open the classroom door a little or when I swing it wide open?” (It is closer when the door is open just a little.)
- “An acute triangle has all three angle measures less than 90 degrees. An obtuse triangle has one angle that is greater than 90 degrees. Imagine an acute triangle with sides of length 4, 5, and 6. Then, imagine an obtuse triangle with sides of length 4 and 5. Does the third side of the obtuse triangle need to be greater than 6?” (It depends on whether the obtuse triangle is between the sides with the given lengths or not. If the obtuse angle is between the sides of length 4 and 5, then the answer is yes. If the obtuse angle is one of the other 2 angles, then the answer is no.)
Tell students that the side of a triangle that is opposite the largest angle will always be the longest, and the side of a triangle that is opposite the smallest angle will always be the shortest.
15.2
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
In this activity, students examine the converse of the Pythagorean Theorem: If the sides of a triangle are of lengths and and , then the triangle is a right triangle. Students do not need to create a rigorous proof of this fact, but the Are You Ready For More? section provides an outline of how the fact can be proven.
Students must make use of the structure of triangles and the relationship between an angle and its opposite side to describe the triangles (MP7).
15.3
Activity
Optional
Instructional Routines
NoneMaterials
NoneActivity Narrative
In this activity, students explore triangles that have side lengths and to arrive at the triangle inequality: . A proof of the inequality is not included here, but students should be able to make sense of the inequality using their intuition about triangles.
Lesson Synthesis
The purpose of this discussion is for students to understand the relationship between the side lengths of a triangle and how to classify a triangle by its angles.
Ask students,
- “What is the converse of the Pythagorean Theorem?” (If a triangle has side lengths and and , then it is a right triangle.)
- “How can you use the side lengths of a triangle to classify the triangle as acute, right, or obtuse based on its angles?” (Label the triangle with sides and with the longest side labeled as . If , then it is a right triangle. If , then it is an acute triangle. If it is an obtuse triangle.)
- “If you know two side lengths of a triangle, what are the maximum and minimum values for the length of the third side?” (The third side must be less than the sum of the other two sides and greater than the positive difference of the other two sides.)
Student Lesson Summary
The Pythagorean Theorem describes a relationship among the side lengths of a right triangle. In particular, if the sides are labeled and with , as the longest side, then
The converse of this theorem is also true. In fact, if we know the lengths of all three sides, we can classify the triangle as either acute, right, or obtuse.
- If , then the triangle has a right angle.
- If , then the triangle has an obtuse angle.
- If , then the triangle has only acute angles.
A key reason this is true is due to the relationship between an angle and the side opposite the angle in a triangle. If the two segments making up the angle are constant, then when the angle is made greater, the opposite side must also be greater.
Another relationship among the sides of a triangle is due to limitations from the ways the ends of the segments can be connected. The length of any side of a triangle must be less than the sum of the other two side lengths in the triangle and must be greater than the positive difference between the other two side lengths. For a triangle with side lengths and , this can be written as