1) This problems has two parts, Part A and Part B. Part A: What is the summation notation of LaTeX: \left(x+y\right)^5 ( x + y ) 5 ? Part B: Expand LaTeX: \left(x+y\right)^5 ( x + y ) 5 using the Binomial Theorem. A: LaTeX: \sum\begin{matrix}5\\k=0\end{matrix }\:\binom{5}{k}x^{5-k}y^k=\binom{5} {0}x^{5-0}y^0+\binom{5}{1}x^{5-1}y^ 1+\binom{5}{2}x^{5-2}y^2+\binom{5}{ 3}x^{5-3}y^3+\binom{5}{4}x^{5-4}y^4 +\binom{5}{5}x^{5-5}y^5 A: LaTeX: \sum\begin{matrix}5\\k=1\end{matrix }\:\binom{5}{k}x^{5-k}y^k=\binom{5} {1}x^{5-1}y^1+\binom{5}{2}x^{5-2}y^ 2+\binom{5}{3}x^{5-3}y^3+\binom{5}{ 4}x^{5-4}y^4+\binom{5}{5}x^{5-5}y^5 A: LaTeX: \sum\begin{matrix}5\\k=0\end{matrix }\:\binom{5}{k}x^{5-k}y^k=\binom{5} {0}x^0y^{5-0}+\binom{5}{1}x^1y^{5-1 }+\binom{5}{2}x^2y^{5-2}+\binom{5}{ 3}x^3y^{5-3}+\binom{5}{4}x^4y^{5-4} +\binom{5}{5}x^5y^{5-5} B: LaTeX: x^5-5x^4y-10x^3y^2+10x^2y^3-5xy^4-y ^5 B: LaTeX: x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+y ^5 B: LaTeX: x^5+5x^4y+10x^3y^2+15x^2y^3+5xy^4+y ^5

Answer: question no. 1 -a. yes, the sides are in the ratio 2 : 5.question no. 2 -b. 59°question no. 3 -c. angles x and y are remote interior angles.question no. 4 -d. 90°, 45° and 45°question no. 5 -b. mn and qrstep-by-step explanation: question no. 1 -rs : xu = 3 : 7.5 = 2 : 5rt : x? = 6.4 : 16 = 2 : 5st : u? = 6 : 15 = 2 : 5as the ratios of all sides are equivalent, the triangles must be congruent and the sides must be proportional.therefore, the answer is: a. yes, the sides are in the ratio 2 : 5.question no. 2 -in the second diagram, angle b is the only letter not shown. therefore, the number must be equivalent to angle b.if we substitute the value of angle b as being 22° in the first diagram, we can now use the angle sum of a triangle to obtain the value of angle a.angle sum of a triangle = 180°therefore: let angle a = aa + 99 + 22 = 180a = 180 - 99 - 22a = 59°therefore, the answer is: b. 59°question no. 3 -angles x and y don't share the vertex of the triangle with the exterior angle w. remote interior angles don't share a vertex of a triangle with an exterior angle. this means that angles x and y are remote interior angles.therefore, the answer is: c. angles x and y are remote interior angles.question no. 4 -the sum of all interior angles of a triangle must be equivalent to 180°.90°, 45° and 45° are the only set of angle measures that are equivalent to 180°.therefore, the answer is: d. 90°, 45° and 45°question no. 5 -the only pair of sides stated in the options which are corresponding are mn and qr.therefore, the answer is: b. mn and qr.

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