Two triangles are similar if their corresponding angles are equal and corresponding sides are: — Proportional.Similar triangles have equal corresponding angles AND proportional corresponding sides (not necessarily equal).
The AA similarity criterion requires how many pairs of equal angles? — 2.AA (Angle-Angle) requires only two pairs of equal angles. The third pair is automatically equal since angles in a triangle sum to 180°.
If △ABC ~ △DEF, what is the ratio AB/DE called? — The scale factor.The scale factor is the ratio of corresponding side lengths in similar figures: k = AB/DE = BC/EF = AC/DF.
If △ABC ~ △DEF with scale factor 2, and the perimeter of △ABC is 30, what is the perimeter of △DEF? — 15.Perimeters scale with the scale factor. Scale factor △ABC to △DEF is 2, so △DEF perimeter = 30/2 = 15.
In △ABC ~ △DEF, ∠A = 40°, ∠B = 80°. What is ∠F? — 60°.∠C = 180° − 40° − 80° = 60°. In similar triangles, ∠C corresponds to ∠F, so ∠F = 60°.
SAS similarity requires two pairs of proportional sides and: — The included angle equal.SAS (Side-Angle-Side) similarity requires two pairs of proportional sides AND the angle BETWEEN those sides (included angle) to be equal.
If △ABC ~ △DEF with AB = 8, DE = 12, and BC = 10, what is EF? — 15.AB/DE = BC/EF → 8/12 = 10/EF → EF = 10 × 12/8 = 15.
Are all equilateral triangles similar to each other? — Yes, because all angles are 60°.All equilateral triangles have three 60° angles. By AA, any two equilateral triangles are similar (regardless of size).
The Triangle Midsegment Theorem says the midsegment is parallel to the third side and its length is: — Half the third side.The midsegment connecting midpoints of two sides is always parallel to the third side and exactly half its length.
How many midsegments does a triangle have? — 3.Each triangle has three midsegments — one connecting each pair of side midpoints.
In △ABC, M is the midpoint of AB and N is the midpoint of AC. If BC = 20, what is MN? — 10.By the Midsegment Theorem, MN = ½ BC = ½ × 20 = 10.
What is the midpoint formula for points (x₁, y₁) and (x₂, y₂)? — ((x₁+x₂)/2, (y₁+y₂)/2).The midpoint of a segment is the average of the x-coordinates and the average of the y-coordinates: M = ((x₁+x₂)/2, (y₁+y₂)/2).
If the midsegment of a triangle is parallel to the base, what angle relationship does this create? — Corresponding angles equal (alternate angles).Parallel lines cut by a transversal create equal corresponding (and alternate) angles. MN ∥ BC means ∠ANM = ∠ABC (corresponding angles).
The three midsegments of a triangle divide it into how many smaller congruent triangles? — 4.The three midsegments divide the original triangle into 4 congruent triangles (the medial triangle and three corner triangles).
In △ABC, midsegment DE connects midpoints of AB and BC. Which side is DE parallel to? — AC.The midsegment connecting midpoints of AB and BC is parallel to the third side, AC.
In △XYZ with midsegment MN (M on XY, N on XZ), if MN = 8, find YZ. — 16.Midsegment = ½ × (third side) → 8 = ½ × YZ → YZ = 16.
SOH-CAH-TOA: what does SOH stand for? — Sin = Opposite/Hypotenuse.SOH stands for Sin = Opposite/Hypotenuse. It is a mnemonic for the three basic trig ratios.
In a right triangle with angle θ, the cosine of θ is defined as: — adjacent/hypotenuse.cos θ = adjacent/hypotenuse (CAH in SOH-CAH-TOA).
In a right triangle with angle θ, the tangent of θ is defined as: — opposite/adjacent.tan θ = opposite/adjacent (TOA in SOH-CAH-TOA).
Which side of a right triangle is always the longest? — The hypotenuse.The hypotenuse is opposite the right angle (90°) and is always the longest side of a right triangle.
If sin θ = 3/5, what is the hypotenuse if the opposite side is 9? — 15.sin θ = opposite/hypotenuse = 3/5. If opposite = 9, then 9/hyp = 3/5 → hyp = 15.
In right △ABC with ∠C = 90° and ∠A = 30°, which expression gives the side BC (opposite to ∠A) if hypotenuse AB = 10? — BC = 10 sin 30°.sin 30° = BC/AB = BC/10 → BC = 10 sin 30° = 10(0.5) = 5.
The Pythagorean Identity states: — sin²θ + cos²θ = 1.The fundamental Pythagorean identity is sin²θ + cos²θ = 1, derived from the Pythagorean theorem applied to a unit circle.
What does the inverse function sin⁻¹(x) find? — The angle whose sine is x.sin⁻¹(x) (also written arcsin x) finds the angle θ such that sin θ = x. It "undoes" the sine function.
In a 45-45-90 triangle, the sides are in ratio: — 1 : 1 : √2.The 45-45-90 triangle has legs in ratio 1:1 and hypotenuse = leg × √2, giving ratio x : x : x√2.
In a 30-60-90 triangle, the sides are in ratio: — 1 : √3 : 2.The 30-60-90 triangle has sides x (short leg, opposite 30°), x√3 (long leg, opposite 60°), and 2x (hypotenuse, opposite 90°).
In a 45-45-90 triangle with leg = 7, what is the hypotenuse? — 7√2.Hypotenuse = leg × √2 = 7√2.
In a 30-60-90 triangle with short leg x = 5, what is the hypotenuse? — 10.Hypotenuse = 2x = 2(5) = 10.
In a 30-60-90 triangle with short leg = 6, what is the long leg? — 6√3.Long leg = x√3 = 6√3.
The Law of Sines states: — a/sinA = b/sinB = c/sinC.Law of Sines: a/sinA = b/sinB = c/sinC. Each side divided by the sine of its opposite angle gives the same value.
The Law of Cosines generalizes which theorem to non-right triangles? — Pythagorean Theorem.Law of Cosines: c² = a² + b² − 2ab·cosC. When C = 90°, cosC = 0, and it reduces to a² + b² = c².
When do you use the Law of Sines? — When given AAS or ASA (two angles and a side).Use Law of Sines when you have two angles and any side (AAS or ASA), or the ambiguous SSA case.
When do you use the Law of Cosines? — When given SAS or SSS.Use Law of Cosines for SAS (two sides + included angle) or SSS (all three sides). It avoids the ambiguous SSA case.
If two angles of a triangle are known, how can you find the third? — Subtract the sum from 180°.Triangle angle sum = 180°. Third angle = 180° − A − B.
After finding all three sides using Law of Cosines, the best way to find remaining angles is: — Use Law of Sines (simpler arithmetic).After finding all sides with the Law of Cosines, switching to the Law of Sines for angles requires simpler arithmetic.
The Law of Cosines is a generalization of which relationship? — The Pythagorean Theorem for non-right triangles.The Law of Cosines c² = a² + b² − 2ab·cosC reduces to the Pythagorean Theorem when the angle C = 90°.