HELP!!!brainlist Drag a reason to each box to complete the flowchart proof.

Mathematics

2 answers:

exis [7]3 years ago

3 0

The complete proof statement and reason for the required proof is as follows:

Statement Reason

m<PNO = 45 Given

MO Given

<MNP and <PNO are a
linear pair of angles Definition of linear pairs of angles

<MNP and <PNO are
supplementary angles Linear Pair Postulate

m<MNP + m<PNO = 180° Definition of supplementary angles

m<MNP + 45° = 180° Substitution property of equality

m<MNP = 135° Subtraction property of equality

Nataly [62]3 years ago

3 0

Answer:

Definition of Linear Pairs of Angles; and Definition of supplementary angles.

Step-by-step explanation:

A linear pair is two angles that form a straight line. Thus our second step, that ∠MNP and ∠PNO are a linear pair, is from the definition of linear pairs.

We know from our third step that linear pairs are supplementary. By the definition of supplementary angles, this means they have a sum of 180°.

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fgiga [73]

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2 years ago

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Calculate the total area of the shaded region.

LUCKY_DIMON [66]

so hmmm seemingly the graphs meet at -2 and +2 and 0, let's check

$\stackrel{f(x)}{2x^3-x^2-5x}~~ = ~~\stackrel{g(x)}{-x^2+3x}\implies 2x^3-5x=3x\implies 2x^3-8x=0 \\\\\\ 2x(x^2-4)=0\implies x^2=4\implies x=\pm\sqrt{4}\implies x= \begin{cases} 0\\ \pm 2 \end{cases}$

so f(x) = g(x) at those points, so let's take the integral of the top - bottom functions for both intervals, namely f(x) - g(x) from -2 to 0 and g(x) - f(x) from 0 to +2.

$\stackrel{f(x)}{2x^3-x^2-5x}~~ - ~~[\stackrel{g(x)}{-x^2+3x}]\implies 2x^3-x^2-5x+x^2-3x \\\\\\ 2x^3-8x\implies 2(x^3-4x)\implies \displaystyle 2\int\limits_{-2}^{0} (x^3-4x)dx \implies 2\left[ \cfrac{x^4}{4}-2x^2 \right]_{-2}^{0}\implies \boxed{8} \\\\[-0.35em] ~\dotfill$

$\stackrel{g(x)}{-x^2+3x}~~ - ~~[\stackrel{f(x)}{2x^3-x^2-5x}]\implies -x^2+3x-2x^3+x^2+5x \\\\\\ -2x^3+8x\implies 2(-x^3+4x) \\\\\\ \displaystyle 2\int\limits_{0}^{2} (-x^3+4x)dx \implies 2\left[ -\cfrac{x^4}{4}+2x^2 \right]_{0}^{2}\implies \boxed{8} ~\hfill \boxed{\stackrel{\textit{total area}}{8~~ + ~~8~~ = ~~16}}$