Angles, Triangles, and Circles

• Solve problems involving supplementary, complementary, and vertical angles
• Apply the triangle angle sum (180°) and the exterior angle theorem, and find a triangle's area with 1/2 × base × height
• Use the Pythagorean theorem and recognize special right triangles (30-60-90, 45-45-90)
• Calculate the area and circumference of a circle, moving between radius, diameter, circumference, and area, and find arc length and sector area as a fraction of the circle
• Finish on one ACT-hard question and say what made it hard

The ACT does not provide a geometry formula sheet, so you must memorize key formulas. For triangles: area = 1/2 × base × height; angles sum to 180°; the Pythagorean theorem a² + b² = c² applies to right triangles. Special right triangles have fixed side ratios: 45-45-90 has sides 1:1:√2 and 30-60-90 has sides 1:√3:2. For circles: area = πr² and circumference = 2πr (or πd); many questions give one of these and ask for another, so find the radius first. Vertical angles (the opposite angles where two lines cross) are equal; supplementary angles sum to 180° and complementary angles sum to 90°. Volume and surface area of 3D solids are not part of today's lesson; they have their own lesson (3D Solids), so today only note that the ACT gives no formula sheet for them either. Arc length and sector area are the same fraction of the circle: a central angle of θ degrees gives arc length (θ/360) × 2πr and sector area (θ/360) × πr², so a 72° sector of a circle with radius 10 has area (72/360) × 100π = 20π. The areas of parallelograms and trapezoids are in the Circles, Polygons, and Area lesson. Estimate before computing: the act's directions say figures are not necessarily drawn to scale, so the look of a figure is only a hint, but a quick estimate from the given numbers still eliminates impossible choices (an arc cannot be longer than the whole circumference, and a side of a triangle is shorter than the other two sides combined). For an odd shape, split it into rectangles, triangles, and circle pieces, or take a whole shape and subtract the missing piece. Chase angles with the straight-line and triangle sums of 180° before reaching for trigonometry. Plane geometry is a steady part of ACT Math, so these facts are worth knowing cold. The hardest questions, at the end of the section: a circle combined with a polygon, such as a right triangle inscribed in a circle, whose hypotenuse is a diameter, or a radius that is also the hypotenuse of a right triangle, where one extra fact turns a circle measurement into a triangle measurement. An equilateral triangle inscribed in a circle cuts the circle into three equal 120° arcs, and a rectangle split into triangles often hides Pythagorean triples (9-12-15, 15-20-25) that share a side.

• Draw and label diagrams whenever the problem describes a geometric figure. Visual aids prevent errors.
• For 'which must be true' questions, test extreme cases to eliminate wrong answers.
• Memorize the 3-4-5, 5-12-13, and 8-15-17 Pythagorean triples. They save calculation time.