Congruent Triangles and Corresponding Parts

13.3: Congruent Triangles and Corresponding Parts. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

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Corresponding parts

Objective: Use the ordered triangle names to match vertices and sides.

Prerequisite skills: triangle correspondence and angle-side relationships. In △ABC≅△DEF, the order pairs A↔D, B↔E, C↔F. Corresponding sides and angles are equal. Names encode correspondence even when the triangles are rotated or reflected. Common error: matching sides solely by their position in a drawing. Restrictions and interpretation: Congruence requires sufficient information: SSS, SAS with included angle, ASA/AAS, or right-triangle HL; AAA proves only similarity and SSA is not generally sufficient.

Formulas

$△ABC≅△DEF ⇒ AB=DE, BC=EF, AC=DF$

Worked examples

Try itGiven △ABC ≅ △DEF and BC = 26 centimeters, what is EF in centimeters?Show answer

The naming order pairs B with E and C with F, so side BC corresponds to EF. Congruent corresponding sides are equal: EF=26 cm. Therefore the requested answer is 26.

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Corresponding parts

Objective: Use the ordered triangle names to match vertices and sides.

Formulas

$△ABC≅△DEF ⇒ AB=DE, BC=EF, AC=DF$

Worked examples

Try itGiven △ABC ≅ △DEF and BC = 26 centimeters, what is EF in centimeters?Show answer

The naming order pairs B with E and C with F, so side BC corresponds to EF. Congruent corresponding sides are equal: EF=26 cm. Therefore the requested answer is 26.

Sufficiency of congruence information

Objective: Distinguish sufficient congruence criteria from angle-only similarity.

SSS fixes all sides; SAS fixes two sides and their included angle; ASA/AAS fixes two angles and one side. In right triangles, matching hypotenuse and one leg is sufficient. Three matching angles alone permit different scales. Common error: using AAA to prove congruence or using an unqualified SSA condition. Restrictions and interpretation: Congruence requires sufficient information: SSS, SAS with included angle, ASA/AAS, or right-triangle HL; AAA proves only similarity and SSA is not generally sufficient.

Formulas

SSS, SAS, ASA, AAS, or right-triangle HL

Worked examples

Try itOne triangle has two sides 26 and 28 with included angle 52°. Which matching data are sufficient for another triangle to be congruent?Show answer

The two sides and their included angle meet SAS. They fix the triangle’s shape and size. Angles alone give similarity; one side or a perimeter does not locate the remaining vertices uniquely. Therefore the requested answer is The same two sides and their included angle.

Algebra in corresponding sides

Objective: Use a congruence equality to find a length.

After proving or being given congruence, set expressions for corresponding sides equal. Solve for the variable, then compute the side length requested. A valid algebraic value must produce positive lengths. Common error: reporting x instead of evaluating the requested side. Restrictions and interpretation: Congruence requires sufficient information: SSS, SAS with included angle, ASA/AAS, or right-triangle HL; AAA proves only similarity and SSA is not generally sufficient. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

corresponding side expressions are equal

Worked examples

Try itTriangles ABC and DEF are congruent in that order. AB = 3x + 2 and DE = 2x + 28. What is AB?Show answer

AB corresponds to DE, so 3x+2=2x+28. Subtract 2x and 2: x=26. Then AB=3(26)+2=80, a positive length. Therefore the requested answer is 80.

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