Warm-UpWarm-Up

If m<J + m<E + m<R = 180°, then construct <R.

4.4 Prove Triangles Congruent by SAS and HL4.4 Prove Triangles Congruent by SAS and HL4.5 Prove Triangles Congruent by ASA and 4.5 Prove Triangles Congruent by ASA and AASAAS

Objectives:

1. To discover and use shortcuts for showing that two triangles are congruent

Congruent Triangles Congruent Triangles (CPCTC)(CPCTC)Two triangles are congruent triangles congruent triangles if

and only if the ccorresponding pparts of those ccongruent ttriangles are ccongruent.

• Corresponding sides are congruent

• Corresponding angles are congruent

Congruent TrianglesCongruent Triangles

Checking to see if 3 pairs of corresponding sides are congruent and then to see if 3 pairs of corresponding angles are congruent makes a total of SIX pairs of things, which is a lot! Surely there’s a shorter way!

Congruence Shortcuts?Congruence Shortcuts?

• Will one pair of congruent sides be sufficient? One pair of angles?

Congruence Shortcuts?Congruence Shortcuts?

• Will two congruent parts be sufficient?

Congruent Shortcuts?Congruent Shortcuts?

• Will three congruent parts be sufficient?

Congruent Shortcuts?Congruent Shortcuts?

• Will three congruent parts be sufficient?

Included Angle Included Side

Congruent Shortcuts?Congruent Shortcuts?

• Will three congruent parts be sufficient?

Investigation: ShortcutsInvestigation: Shortcuts

Well, we know that SSS is a valid shortcut, and I’ll give you the hint that 2 others in the list do not work.

We will test the remaining 5 in class. For each of these, you will be given three pieces to form a triangle. If the shortcut works, one and only one triangle can be made with those parts.

Shortcuts?:SSSSSASASASAAASAAA

√√

Copying an AngleCopying an Angle

5. Put point of compass on B and pencil on C. Make a small arc.

Congruence ShortcutsCongruence Shortcuts

Side-Side-Side (SSS) Congruence Postulate:Side-Side-Side (SSS) Congruence Postulate:If the three sides of one triangle are congruent to

the three sides of another triangle, then the two triangles are congruent.

Congruence ShortcutsCongruence Shortcuts

Side-Angle-Side (SAS) Congruence Postulate:Side-Angle-Side (SAS) Congruence Postulate:

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Congruence ShortcutsCongruence Shortcuts

Angle-Side-Angle (ASA) Congruence Postulate:Angle-Side-Angle (ASA) Congruence Postulate:

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

Congruence ShortcutsCongruence Shortcuts

Angle-Angle-Side (AAS) Congruence Theorem:Angle-Angle-Side (AAS) Congruence Theorem:If two angles and a non-included side of one

triangle are congruent to the corresponding two angles and the non-included side of another triangle, then the two triangles are congruent.

And One More!And One More!

Hypotenuse-Leg (HL) Congruence Theorem:Hypotenuse-Leg (HL) Congruence Theorem:

If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and leg of another right triangle, then the two triangles are congruent.

Example 1Example 1

What is the length of the missing leg in the each of the right triangles shown?

5 cm

5 cm

13 cm

13 cm

Notice that the pieces given here correspond to SSA, which doesn’t work. Because of the Pythagorean Theorem, right triangles are an exception.Therefore, rt. triangles have theorems such as HL (hypotenuse-leg) and LL (leg-leg)

12

12

Example 2Example 2

Determine whether the triangles are congruent in each pair.

Yes, SASNo

Example 3Example 3

Determine whether the triangles are congruent in each pair. Answer and explain

which theorem in your notebook

Example 4Example 4

Explain the difference between the ASA and AAS congruence shortcuts.

Answer in your notebook.

Example 5Example 5

TRY IT in your notebook!I will pick someone at random to work it on the board Ain’t life GRAND!