Lesson 2: Drawing Polygons with Given Conditions

Property

If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.
If $\overline{AB} \cong \overline{DE}$, $\overline{BC} \cong \overline{EF}$, and $\overline{AC} \cong \overline{DF}$, then $\Delta ABC \cong \Delta DEF$.

Examples

• Construction: If you are given three specific stick lengths (e.g., 5 cm, 7 cm, and 9 cm), you can only build one unique triangle. Anyone else using those same three lengths will build an exact clone of your triangle.
• Structural Engineering: If you use three beams with lengths of 6 feet, 8 feet, and 9 feet to build a roof support, the resulting triangle will always hold the exact same shape, making it incredibly stable.
• Proof Context: If diagram markings show all three pairs of corresponding sides have matching tick marks, you can immediately declare the triangles congruent by the SSS Postulate.

Explanation

Property

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.

If $\overline{AB} \cong \overline{DE}$, $\angle B \cong \angle E$, and $\overline{BC} \cong \overline{EF}$, then $\Delta ABC \cong \Delta DEF$.

Examples

• Carpentry: A carpenter joins a 10-foot beam and a 12-foot beam exactly at a 45° angle. By locking in those two lengths and the angle between them, the distance between their free ends is automatically determined, locking in the third side and creating a unique triangle.
• Robotics: If a robot arm has a 40 cm segment and a 60 cm segment joined by a motorized "elbow" set to a 110° angle, the position of the arm's tip is uniquely determined.
• Proof Context: In two triangles, if you have matching sides of 6 cm and 8 cm, and the angle directly between them is 50° in both figures, the triangles are congruent by SAS.