ACT Preparation / Unit 13 ACT 1230 review, 2 to 3 minutes
One Figure, Five Facts
The idea
Plane geometry on the test runs on a short list of facts. Angles on a line add to 180°, and vertical angles are equal; with parallel lines and a transversal, alternate interior and corresponding angles are equal, and same-side interior angles add to 180°. A triangle's angles add to 180°, an exterior angle equals the two remote interior angles, and an isosceles triangle has equal base angles. Congruent triangles are proved by SSS, SAS, ASA, AAS or hypotenuse-leg, never by SSA or AAA. Parallelograms have equal opposite sides and angles and bisecting diagonals. Similar triangles, proved by AA, have proportional sides, and a scale factor k multiplies areas by k² and volumes by k³. In a right triangle, a² + b² = c², and the special triangles have sides in the ratios 1 : 1 : √2 and 1 : √3 : 2.
In practice
Worked example: two parallel streets, 300 meters apart, are crossed by a straight diagonal road that meets the northern street at an angle of 60°. A footpath runs from point A, where the road leaves the northern street, straight south at right angles to both streets, to point B on the southern street. The road reaches the southern street at C. At C, the road makes a 60° angle with the southern street, by alternate interior angles. Triangle ABC has a right angle at B and 60° at C, so the angle at A is 30°: a 30-60-90 triangle. The path AB, 300 meters, faces the 60° angle, so it is the longer leg. The shorter leg BC is 300/√3 = 100√3, about 173 meters, and the road AC, the hypotenuse, is 200√3, about 346 meters.
Where people slip
Three habits carry forward. Name the fact behind every step, because each move in a figure uses exactly one. Match sides by the angles they face, in similar and special triangles alike. And trust the given measures over the drawing. The next unit measures figures: perimeters, areas, volumes and circles, and the trigonometry that links a right triangle's sides to its angles.
Your place on the trail
Every lesson rests on earlier ones and opens later ones. Tap any to jump.
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