Triangle abc is shown below.
triangle abc; line passes through points d, b, and e
given: ∆abc
prove: all three angles of ∆abc add up to 180°.
the flow chart with missing reason proves the measures of the interior angles of ∆abc total 180°.
which reason can be used to fill in the numbered blank space?
associative property of addition
triangle exterior angle theorem
angle addition postulate
commutative property of addition

The correct answer (though it is not one of the choices) is:

Alternate Interior Angles Theorem.

Explanation:

Alternate interior angles are angles that are inside the parallel lines and on opposite sides of the transversal.  Treating BA as a transversal, ∠DBA and ∠BAC are both inside the parallel lines and on opposite sides of BA, the transversal.  This makes them alternate interior angles, and means that they are congruent.

answer: ratio of volume of cone (with 3 times the height and equal diameter) to volume of cylinder is 2 to 1. the value of π is not used.

step-by-step explanation:

radius is diameter/2 = 4.

area of circle is π r^2.

volume of cylinder is height × area of base = hπr^2 = 3(4^2) π = 48π cubic inches.

not knowing the formula for volume of cone, i calculate it using calculus (which is for calculating things).

volume of cone is integral 0 to h area(y) dy. area(y) is π r(y)^2. r(y) runs r to zero as y runs zero to h, r(0) = r, r(h) = 0.

r(y) = r - r/h y

area(y) = π (r - r/h y)^2 = π (r^2 - 2r^2y/h + r^2y^2/h^2) = π r^2(1-2y/h+y^2/h^2)

antiderivative area(y) = π r^2 (y-y^2/h+y^3/(3h^2))

integral 0 to h = π r^2(h-h^2/h+h^3/(3h^2) - (0)) = π r^2(h - h + h/3) = π r^2(h/3) = π/3 r^2 h

volume of this cone is π/3(4^2)18 cubic inches.

= 6 16 π = 96π cubic inches.

cone has twice the volume, 96/48.

cone height is 6 times the cylinder.

ratio of volume formula: π r^2 cancels.

cone over cylinder ratio is 6h/3 to h, 2 to 1.