Proving Triangle Congruence

5.3 Proving Triangle Congruence by SAS Theorem 5.5 Side-Angle-Side (SAS) Congruence Theorem If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.   If AB ≅ DE, ∠A ≅ ∠D, and AC ≅ DF, then △ABC ≅ △DEF   [Diagram of two triangles ABC and DEF, with corresponding sides and angles marked as congruent] Extra Practice In Exercises 1 and 2, write a proof. 1. Given: BD ⊥ AC, AD ≅ CD Prove: △ABD ≅ △CBD STATEMENTS REASONS BD ⊥ AC Given ∠BDA = 90° definition of Perpendicular ∠BDC = 90° definition of Perpendicular ∠BDA ≅ ∠BDC Substitution AD ≅ CD Given BD ≅ BD Reflexive △ABD ≅ △CBD by SAS (according to the picture layout) or you could write the proof STATEMENTS REASONS BD ≅ DB Symmetric BD ⊥ AC Given ∠BDA = 90° definition of ⊥ ∠BDC = 90° definition of ⊥ ∠BDA ≅ ∠BDC Substitution AD ≅ CD Given △ABD ≅ △CBD by SAS 5.3 2. Given: JN ≅ MN, NK ≅ NL Prove: △JNK ≅ △MNL STATEMENTS REASONS JN ≅ MN Given NK ≅ NL Given ∠JNK ≅ ∠MNL Vertical Angles △JNK ≅ △MNL by SAS In Exercises 3 and 4, use the given information to name two triangles that are congruent. Explain your reasoning. 3. 1 ∠EPF ≅ ∠GPH, and P is the center of the circle.   1. ca01001129.schoolwires.net ca01001129.schoolwires.net [Diagram of a circle with points E, F, G, H, and P on the circumference] Since ∠EPF ≅ ∠GPH and EP ≅ FP ≅ HP ≅ GP (radii of circles are the same) △EPF ≅ △GPH by SAS 4. ABCDEF is a regular hexagon. [Diagram of a regular hexagon] All sides are ≅ All angles are ≅ So △ABC ≅ △BCD ≅ △CDE ≅ △DEF ≅ △EFA ≅ △FAB by SAS 5. A quilt is made of triangles. You know ∥ and PS ≅ QR. Show that △PQR ≅ △RSP [Diagram of a quilt with triangles] STATEMENTS REASONS PS ∥ QR Given ∠RSP ≅ ∠PRQ Alternate Interior Angles (since we have Parallel lines) PS ≅ QR Given PR ≅ RP Symmetric △PQR ≅ △RSP by SAS