Lesson 5 Congruent Triangles to the Rescue Practice Understanding
Previously, we have conjectured that the opposite sides of a parallelogram are congruent, based on experimentation and an explanation using rigid transformations.
Based on this assumption, mark congruent segments in the diagram.
Would drawing one of the diagonals of the parallelogram provided in problem 1 divide the parallelogram into two congruent triangles?
If so, what triangle congruence criterion did you use to decide that the two triangles are congruent?
Name a pair of congruent angles in the diagram provided in problem 1.
How do you know those angles are congruent?
Identify congruent triangles, and write congruency statements.
Use triangle congruence criteria to justify other properties of geometric figures.
How can I capture what I see when I identify congruent triangles in symbolic notation?
How do I use the triangle congruence criteria to justify other properties of geometric figures?
When might it be helpful to decompose a geometric figure into triangles so that I can use triangle congruence criteria to prove something else about the figure?
Open Up the Math: Launch, Explore, Discuss
Zac and Sione are exploring isosceles triangles—triangles in which two sides are congruent:
Zac: “I think every isosceles triangle has a line of symmetry that passes through the vertex point of the angle made up by the two congruent sides and the midpoint of the third side.”
Sione: “That’s a pretty big claim—to say you know something about every isosceles triangle. Maybe you just haven’t thought about the ones for which the claim isn’t true.”
Zac: “But I’ve folded lots of isosceles triangles in half, and it always seems to work.”
Sione: “Lots of isosceles triangles are not all isosceles triangles, so I’m still not sure.”
What do you think about Zac’s claim? Do you think every isosceles triangle has a line of symmetry? If so, what convinces you this is true? If not, what concerns do you have about his statement?
What else would Zac need to know about the crease line in order to know that it is a line of symmetry? (Hint: Think about the definition of a line of reflection.)
Sione thinks Zac’s “crease line” (the line formed by folding the isosceles triangle in half) creates two congruent triangles inside the isosceles triangle. Which criterion—ASA, SAS, or SSS—could he use to support this claim? Describe the sides and/or angles you think are congruent, and explain how you know they are congruent.
If the two triangles created by folding an isosceles triangle in half are congruent, what does that imply about the “base angles” of an isosceles triangle (the two angles that are not formed by the two congruent sides)?
If the two triangles created by folding an isosceles triangle in half are congruent, what does that imply about the “crease line”? (You might be able to make a couple of claims about this line—one claim comes from focusing on where the line meets the third, noncongruent side of the triangle; a second claim comes from focusing on where the line intersects with the vertex angle formed by the two congruent sides.)
Like Zac, you have done some experimenting with lines of symmetry, as well as with rotational symmetry. In the tasks you did in Symmetries of Quadrilaterals and Quadrilaterals—Beyond Definition, you made some observations about sides, angles, and diagonals of various types of quadrilaterals, based on your experiments and knowledge about transformations. Many of these observations can be further justified based on looking for congruent triangles and their corresponding parts, just as Zac and Sione did in their work with isosceles triangles.
Pick one of the following quadrilaterals to explore:
A rhombus is a quadrilateral in which all sides are congruent.
A rectangle is a quadrilateral that contains four right angles.
A square is both a rectangle and a rhombus, that is, it contains four right angles and all sides are congruent.
Draw an example of your selected quadrilateral with its diagonals. Label the vertices of the quadrilateral
Based on (1) your drawing, (2) the given definition of your quadrilateral, and (3) information about sides and angles that you can gather based on lines of reflection and rotational symmetry, list as many pairs of congruent triangles as you can find.
For each pair of congruent triangles that you list, state the criterion you used—ASA, SAS, or SSS–to determine that the two triangles are congruent, and explain how you know that the angles and/or sides required by the criterion are congruent (see the chart).
Criteria Used (ASA, SAS, SSS)
How do I know that the sides and/or angles required by the criteria are congruent?
If I say
Based on SSS
Then I need to explain:
so I can use SSS criteria to say
Ready for More?
We have accepted many attributes of different types of parallelograms based on experimentation with rigid transformations. Now that you have identified some congruent triangles in parallelograms, try to use the congruent triangles to justify something else about the different types of parallelograms, such as:
The diagonals bisect each other.
The diagonals are congruent.
The diagonals are perpendicular to each other.
The diagonals bisect the angles of the quadrilateral.
Pick one of the bulleted statements you think is true about a specific type of parallelogram, and try to write an argument that would convince Zac and Sione that the statement is true.
Today I learned several strategies for working with diagrams to prove statements about congruence, such as:
Adding Notation, Vocabulary, and Conventions
Sometimes it is useful to add an auxiliary line to a diagram.
An auxiliary line is
In this case, the auxiliary line allowed me to
In this lesson, we found several strategies for justifying statements about congruence by first looking for congruent triangles within a geometric figure. One helpful strategy was adding auxiliary lines to the figure. Another was to use the idea that corresponding parts of congruent triangles are congruent.
The figure shows a pair of lines; one is the pre-image and the other is the image.
Define the transformation used to create the image.
Write the equation of each line.
Why is a geometric compass a useful tool in drafting, architecture, and geometry?