Triangles, similarity and polygons — Digital SAT Math practice
Angle sums, similar triangles, polygon angles.
Key terms
- Similar Triangles: Two triangles with identical angles and sides in a constant ratio, preserving shape but not necessarily size.
- Isosceles Triangle: A triangle with at least two equal sides, which forces the opposite angles to be equal as well.
- Remote Interior Angles: The two interior angles of a triangle that do not share a vertex with a given exterior angle.
- Polygon: A closed plane figure bounded by straight line segments, where n represents the number of sides.
Common traps
- Assuming triangles are similar just because they look the same size or shape without verifying angle measures or side ratios.
- Forgetting that the triangle inequality requires every single side to be strictly less than the sum of the other two sides.
- Misaligning corresponding sides when setting up similarity proportions, which flips the ratio and produces incorrect answers.
Key ideas to know
Exterior Angle Theorem
In triangle ABC, angle A is 40 degrees and angle B is 75 degrees. What is the measure of the exterior angle at vertex C?
115 degrees. The exterior angle equals the sum of the two remote interior angles (40 + 75 = 115).
Watch out: Assuming the exterior angle plus the adjacent interior angle equals 90 degrees instead of 180.
Matching Corresponding Sides
Triangle DEF is similar to triangle D'E'F'. If DE = 6, EF = 8, DF = 10, and D'E' = 3, what is the length of E'F'?
4. Set up a proportion using corresponding sides: DE/D'E' = EF/E'F', so 6/3 = 8/x. Solving gives x = 4.
Watch out: Pairing sides based on alphabetical order rather than matching angles to find correct corresponding sides.
Polygon Interior Angle Sum
What is the sum of the interior angles of a regular hexagon?
720 degrees. Use the formula (n - 2) x 180, where n = 6. So (6 - 2) x 180 = 4 x 180 = 720.
Watch out: Forgetting to multiply by 180 or using n instead of n-2 in the formula.
Triangle inequality
A student says sides of length 2, 5 and 8 can form a triangle because 2 + 5 = 7 is close to 8. What is the error?
The two shorter sides must add to more than the longest side. Here 2 + 5 = 7 is less than 8, so no triangle exists.
Watch out: Accepting a sum that is close to, or equal to, the longest side. It must be strictly greater.
Sample questions
Question 1
Triangles ABC and DEF are similar, with vertices corresponding in that order. AB = 11, BC = 12, and EF = 48. What is the length of DE?
- 48
- 45
- 44
- 47
Show answer and explanation
Answer: 44
DE/AB = EF/BC → DE/11 = 48/12 → DE = 44.