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 1-inch strips with evenly-spaced 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 Side-Side-Side, Angle-Side-Angle, and Side-Angle-Side 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 Side-Side-Side, Angle-Side-Angle, and Side-Angle-Side 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 Preparation

For the activities in this lesson and the next section, students will need strips cut from copies of the blackline masters. Prepare enough 1-inch strips with evenly-spaced 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 Side-Side-Side, Angle-Side-Angle, and Side-Angle-Side Triangle Congruence Theorems in proofs.
  • I can write conjectures about quadrilaterals.

CCSS Standards

Building On

Addressing

Building Towards

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 if-then 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.