Lesson 14 of 16
The big picture
Show on a big screenTwo figures, twelve ideas to check them. All on one screen.
| Idea | Remember | Example |
|---|---|---|
| Landing one on the other | Congruent figures land exactly, after any slide, turn or flip. | 5, 6, 7 cm and 5, 6, 8 cm cannot land. |
| Slide, turn, flip | No side or corner changes. | A flipped sleeve still fits. |
| A bigger copy | Same corners, longer sides. Not congruent. | 40, 50, 60 mm becomes 56, 70, 84 mm at 140%. |
| One line, one corner | Compare length or degrees. Tilt does not matter. | Two 60 mm threads match. |
| Three sides (SSS) | Three equal pairs of sides make congruent triangles. | 52, 39 and 68 mm. |
| Two sides and the angle between (SAS) | The angle must sit between the sides. | Move it, and two triangles may fit. |
| Two angles and a side (ASA, AAS) | Find the third angle. Match the side. | 50° and 75° leave 55°. |
| Three angles (AAA) | Same shape. Size stays open. | 40°, 60°, 80° at two sizes. |
| Writing a congruence | In ΔABC ≅ ΔXYZ, the letters match in order. Δ means triangle. ≅ means is congruent to. A matches X, B matches Y, and C matches Z. Matching sides are equal. Matching angles are equal. | ΔPQR ≅ ΔUVT |
| RHS | RHS stands for right angle, hypotenuse, side. Take two right-angled triangles. Suppose their hypotenuses are equal. Suppose one more side is equal. Then they are congruent. The hypotenuse is the side across from the right angle. | Right angle, 13 cm, 5 cm. |
| Equal sides, equal angles | Draw the altitude from the top of an isosceles triangle. An altitude runs from a corner, square to the opposite side. It cuts the triangle into two right-angled triangles. They are congruent by RHS, as shown below. So the angles across from the equal sides are equal. Any two sides of an equilateral triangle are equal. So all three angles are equal. Each is 60°, because they add to 180°. | A top angle of 50° leaves 65° each. |
| Twins in a figure | To prove two triangles congruent, find three marked facts. Match them to SSS, SAS, ASA, AAS or RHS. The facts must fit one test exactly. In SAS, the angle sits between the two sides. A side that two triangles share counts as an equal pair. | A kite: SSS. |
Chapter 1 of Part II is Geometric Twins. Your book asks you to prove twins with these tests. Try its exercises next.
One habit runs through all twelve. Check before you say yes.