Proving Congruence by Transformations

Pre-Algebra · Axiom Academy

LESSON Proving Congruence by Transformations Two figures are congruent exactly when a sequence of rigid motions slides, turns, or flips one onto the other. 1. Congruent Means "Movable Onto" Start with two triangles in different spots. They look the same — but looking isn't proof. The test: can you slide, turn, and flip one until it lands exactly on the other? Watch the blue triangle travel and settle perfectly onto the green one — every vertex meets its partner. Only three moves are allowed, because only these keep every length and angle the same. Watch one shape get slid , turned , and flipped — its size and shape never change. A fourth move, a dilation (resizing), is shown last in red: it breaks congruence, so it is not a rigid motion. Slide every point the same way — no turning, no flipping. Turn the whole figure by a fixed angle about a center point. Flip across a line — a mirror image, same size and shape. 3. Building a Congruence Proof A proof names a specific sequence of rigid motions and shows it lands on . Here has vertices A(1,1) , B(4,1) , C(1,3) , and the target is at D(7,4) , E(10,4) , F(7,2) . The orientation is flipped, so we need a reflection. Watch the two moves play out on the grid: first a reflection over the x -axis , then a translation . Step 1 — reflect over the x -axis: Step 2 — translate right 6, up 5: Reading the map vertex by vertex Every image vertex lands on its target, so . 4. Why It Works: What Stays Fixed

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