Geometry Unit 4: Congruence

Unit Driving Question 

What does it mean for geometric figures to be congruent?

Essential Questions 

1. How do we know when two geometric figures are congruent?  
2. How do we prove a statement is true?  
3. How do different properties define a quadrilateral? What relationships form different classifications within the class of quadrilaterals?  

Launch Task 

Big Ideas for Development Lessons 

Closure & Assessment 

1 Lesson 

4 Weeks (approximately 1 week per big idea) 

1 Week 

Click on the links below to see each Big Idea's Lesson Overview (includes links to teacher notes and student activities) 

1. Polygons are comprised of triangles
2. Congruent polygons are defined by their congruent angles and sides 
3. Congruent corresponding angles and sides are used to prove triangles are congruent
4. Quadrilaterals can be classified by their sides, diagonals and angle measures

1. Formative Assessment 1 (after Big Idea 3) 
2. Re-engagement Activity (Not Provided) 
3. Unit Assessment 

Big Idea 1: Polygons are comprised of triangles

Lessons and Additional Activities

Big Idea 1 Lessons 1-4 Overview (includes links to teacher notes and student activities)

Evidence of Understanding

Explore and prove relationships about interior and exterior angles of polygons

• Apply knowledge about triangles to generate and test conjectures polygons

• Use diagonals to partition figures into triangles to determine the sum of the interior angles of a polygon

• Find the value of missing interior and exterior angles in a polygon

Explore and prove relationships about polygons

• Find the area and perimeter of polygons

Explore and prove relationships of regular polygons

• Apply knowledge about triangles to generate and test conjectures about regular polygons

• Find the value of missing interior and exterior angles in a regular polygon

• Find the area and perimeter of a regular polygon 

Big Idea 2: Congruent polygons are defined by their congruent angles and sides.

Lessons and Additional Activities

Big Idea 2 Lessons 1-4 Overview (includes links to teacher notes and student activities)

Evidence of Understanding 

Describe qualities that make two polygons congruent or incongruent

• Identify corresponding parts (angles and sides) of polygons by annotating

• Use notation to signify congruence

• Use examples and non-examples to justify that corresponding parts of congruent polygons congruent

• Explore whether equal perimeters or areas mean figures are congruent (or vice versa: if figures are congruent then decide if their perimeters or areas are equal)

Big Idea 3: Congruent corresponding angles and sides are used to prove triangles are congruent.

Lessons and Additional Activities

Big Idea 3 Lessons 1-4 Overview (includes links to teacher notes and student activities)

Evidence of Understanding 

Prove two triangles are congruent

• Identify corresponding congruent parts and apply the criteria of SSS, SAS, ASA, or AAS

• Prove two right triangles are congruent when corresponding Hypotenuse-Leg (HL) are congruent

Justify the minimum requirements that show two triangles are congruent

• Make conjectures about the minimum corresponding parts of the triangle needed to construct a congruent triangle

• Experiment with constructions to support these claims

• Give examples, non-examples, or counterexamples about these claims

• Justify that when all corresponding sides of two triangles are congruent (SSS) there is sufficient evidence to show that these two triangles are congruent

• Justify that when two corresponding sides and the included angle are congruent (SAS) there is sufficient evidence to show that these two triangles are congruent

• Justify why the angle has to be the included angle of the corresponding sides (SSA does not work)

• Establish the minimum criteria necessary to prove two right triangles are congruent using the hypotenuse and a leg (HL)

• Justify that when two corresponding angles and the included side of two triangles are congruent (ASA) there is sufficient evidence to show that these two triangles are congruent 

Big Idea 4: Quadrilaterals can be classified by their sides, diagonals, and angle measures. 

Lessons and Additional Activities

Big Idea 4 Lessons 1-3 Overview (includes links to teacher notes and student activities)

Evidence of Understanding 

Distinguish trapezoids, parallelograms, rectangles, kites, rhombuses, and squares using properties of their sides and angles

• Identify and use properties that result in quadrilaterals being part of the same “family”

• Examples: rectangle, square, and rhombus are all parallelograms

Explore and prove relationships about interior and exterior angles of quadrilaterals

• Justify that interior angles of a quadrilateral always add up to 360° using examples and non-examples

Explore and prove relationships about angles and sides of a parallelogram

• Identify and justify congruent angles and sides of a parallelogram

• Prove opposite sides and angles of a parallelogram are congruent

• Apply properties of parallel lines cut by a transversal

• Prove same side/consecutive interior angles of a parallelogram are supplementary

• Prove a given figure is a square, rectangle, or rhombus

• Apply properties of parallelograms, parallel lines, and triangle congruence criteria

• Find the measure of a missing value or measurements

Investigate the relationship between the diagonals of a quadrilateral and its other characteristics

• Prove diagonals of parallelograms bisect each other

• Prove the converse statement (if diagonals bisect each other then it is a parallelogram)

• Prove diagonals of rectangles are congruent

• Investigate diagonals of an isosceles trapezoid and use them show the converse statement is false (if diagonals are congruent, then it is a rectangle)

• Prove diagonals of a rhombus perpendicularly bisect one another

• Demonstrate perpendicular diagonals do not necessarily bisect one another (ex: kites)

• Find the measure of the missing length of a diagonal