Lesson 10
Practicing Proofs
Lesson Narrative
Throughout this unit, students have proved a variety of figures congruent. They started with segments (2 vertices), then triangles (3 vertices), and now are proceeding to the next most complicated geometric structure, quadrilaterals (4 vertices).
In this lesson, students use 1inch strips with evenlyspaced holes and metal paper fasteners to make conjectures about diagonals of quadrilaterals. Then they have a chance to practice using diagrams to recognize when the SideSideSide, AngleSideAngle, and SideAngleSide Triangle Congruence Theorems apply. Finally, students match diagrams with statements about quadrilaterals, and write a proof using the analyses they did earlier.
As students analyze diagrams to see which triangle congruence theorems apply, they look for and make use of structure (MP7). The work of moving from conjectures to proof is another opportunity to look for and make use of structure, as well as construct viable arguments (MP3).
Learning Goals
Teacher Facing
 Generate (orally and in writing) conjectures about quadrilaterals.
 Identify situations where SideSideSide, AngleSideAngle, and SideAngleSide Triangle Congruence Theorems apply, and use them to prove theorems (in writing).
Student Facing
 Let’s practice what we’ve learned about proofs and congruence.
Required Materials
Required Preparation
For the activities in this lesson and the next section, students will need strips cut from copies of the blackline masters. Prepare enough 1inch strips with evenlyspaced holes so that every student gets 2 longer strips and 2 shorter strips. Put each set of strips in an envelope. Prepare one envelope for each student. To prepare the strips and fasteners to be used throughout the unit, include 4 metal fasteners in each envelope.
Learning Targets
Student Facing
 I can use the SideSideSide, AngleSideAngle, and SideAngleSide Triangle Congruence Theorems in proofs.
 I can write conjectures about quadrilaterals.
CCSS Standards
Glossary Entries

auxiliary line
An extra line drawn in a figure to reveal hidden structure.
For example, the line shown in the isosceles triangle is a line of symmetry, and the lines shown in the parallelogram suggest a way of rearranging it into a rectangle.

converse
The converse of an ifthen statement is the statement that interchanges the hypothesis and the conclusion. For example, the converse of "if it's Tuesday, then this must be Belgium" is "if this is Belgium, then it must be Tuesday."

corresponding
For a rigid transformation that takes one figure onto another, a part of the first figure and its image in the second figure are called corresponding parts. We also talk about corresponding parts when we are trying to prove two figures are congruent and set up a correspondence between the parts to see if the parts are congruent.
In the figure, segment \(AB\) corresponds to segment \(DE\), and angle \(BCA\) corresponds to angle \(EFD\).

parallelogram
A quadrilateral in which pairs of opposite sides are parallel.