Triangle Congruence
SSS, SAS, and ASA Postulates
Today's Objective
Do Now: Corresponding Parts
Given that \( \triangle ABC \cong \triangle XYZ \), list the corresponding congruent parts.
\( \angle A \cong \) ?
\( \angle B \cong \) ?
\( \angle C \cong \) ?
\( \overline{AB} \cong \) ?
\( \overline{BC} \cong \) ?
\( \overline{AC} \cong \) ?
\( \angle A \cong \_\_\_\_\_ \) , \( \angle B \cong \_\_\_\_\_ \) , \( \angle C \cong \_\_\_\_\_ \)
\( \overline{AB} \cong \_\_\_\_\_ \) , \( \overline{BC} \cong \_\_\_\_\_ \) , \( \overline{AC} \cong \_\_\_\_\_ \)
Congruent Triangles
Two triangles are congruent if all of their corresponding parts (angles and sides) are congruent.
But... we don't need to check all 6 parts! We can use shortcuts (postulates) to prove triangles are congruent.
Side-Side-Side (SSS)
If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
Side-Angle-Side (SAS)
If two sides and the included angle of one triangle are congruent to the corresponding parts of another, then the triangles are congruent.
The "included angle" is the angle between the two sides.
Angle-Side-Angle (ASA)
If two angles and the included side of one triangle are congruent to the corresponding parts of another, then the triangles are congruent.
The "included side" is the side between the two angles.
How to Prove Triangles are Congruent
Step 1: Mark the information given (tick marks for sides, arcs for angles).
Step 2: Look for "free" information:
- Shared Side (Reflexive Property)
- Vertical Angles (They are always congruent)
Step 3: Count your congruent parts (Sides and Angles).
Step 4: Match your parts to a postulate: SSS, SAS, or ASA.
Practice: Which postulate?
I Do
Given: 3 pairs of congruent sides.
SSS
We Do
Given: Two angles and the side between them.
ASA
You Do
Given: A shared side, and one pair of congruent sides and angles next to the shared side.
SAS
I Do: Given 3 pairs of congruent sides, which postulate proves congruence? SSS
We Do: Given two angles and the included side, which postulate?
You Do: In a diagram with a shared side, one pair of congruent sides, and one pair of congruent included angles, which postulate?
Independent Practice
Are the triangles congruent? If so, state the postulate.
Level 1 Problem:
1. Two triangles are shown. One has sides 5, 6, 7. The other has sides 7, 5, 6.
Level 2 Problem:
2. Two triangles share a common side. The other two sides of each triangle are marked as congruent.
Level 3 Problem:
3. Two triangles are formed by intersecting lines. The vertical angles are congruent. The sides adjacent to the vertical angles are marked congruent.
Independent Practice
1. (Level 1) Triangles with sides 5,6,7 and 7,5,6. Congruent? Postulate?
2. (Level 2) Two triangles share a side, other sides are congruent. Congruent? Postulate?
3. (Level 3) Triangles formed by intersecting lines with vertical angles and adjacent sides congruent. Congruent? Postulate?
Exit Ticket
Can the two triangles be proven congruent with the information given in the diagram? If so, state the postulate.
Diagram shows two triangles. Each has a 40° angle and a 60° angle. The side opposite the 60° angle is marked congruent in both.
No, not enough information. This is AAS, which we haven't learned yet. It is not SSS, SAS, or ASA.