Concept Byte: Exploring AAA and SSA

Use With Lesson 4-3

TECHNOLOGY

So far, you know four ways to conclude that two triangles are congruent—SSS, SAS, ASA, and AAS. It is good mathematics to wonder about the other two possibilities.

Activity 1

Construct Use geometry software to construct A B → eh b vector  and A C → . eh c vector , .  Construct B C ¯ b c bar  to form Δ A B C . cap delta eh b c .  Construct a line parallel to B C ¯ b c bar  that intersects A B → eh b vector  and A C → eh c vector  at points D and E to form Δ A D E . cap delta eh d e .

Investigate Are the three angles of Δ A B C cap delta eh b c  congruent to the three angles of Δ A D E ? cap delta eh d e question mark  Manipulate the figure to change the positions of D E ¯ d e bar  and B C ¯ . b c bar , .  Do the corresponding angles of the triangles remain congruent? Are the two triangles congruent? Can the two triangles be congruent?

Activity 2

Construct Construct A B → . eh b vector , .  Draw a circle with center C that intersects A B → eh b vector  at two points. Construct A C ¯ . eh c bar , .  Construct point E on the circle and construct C E ¯ . c e bar , .

Investigate Move point E around the circle until E is on A B → eh b vector  and forms Δ A C E . cap delta eh c e .  Then move E on the circle to the other point on A B → eh b vector  to form another Δ A C E . cap delta eh c e .

Compare AC, CE, and m ∠ A m angle eh  in the two triangles. Are two sides and a nonincluded angle of one triangle congruent to two sides and a nonincluded angle of the other triangle? Are the triangles congruent? If you change the measure of ∠ A angle eh  and the size of the circle, do you get the same results?

Exercises

1. Make a Conjecture Based on your first investigation above, can you prove triangles congruent using AAA? Explain.

For Exercises 2–4, use what you learned in your second investigation above.

2. Make a Conjecture Can you prove triangles congruent using SSA? Explain.
3. Manipulate the figure so that ∠ A angle eh  is obtuse. Can the circle intersect A B → eh b vector  twice to form two triangles? Would SSA work if the congruent angles are obtuse? Explain.
4. Suppose you are given C E ¯ , A C ¯ , c e bar , comma . eh c bar , comma  and ∠ A . angle eh .  What must be true about CE, AC, and m ∠ A m angle eh  so that you can construct exactly one Δ A C E ? cap delta eh c e question mark  (Hint: Consider cases.)

Table of Contents