A)-5+12=7
б)-43,2+31,2=-12
в)1) 29 : 29/13=29*13/29=13.
2) 13-13,6=-0,6.
3) -0,6+1=0,4.
(13,3: 0,38-32,54):(3,2*0,65+1,32) ; 1)13,3:0,38=35; 2)35-32,54=2,46;
3)3,2*0,65=2,08; 4)2,08+1,32=3,4;
5)2,46:3,4=246/100*10/34=246/340=123/170
2 целых 2/11×1 целую 3/8÷3/4=24/11×11/8×4/3=4
По внешнему виду определить, какие из углов равны, невозможно, если градусные меры углов близки по своим значениям. Нужно вычислять. Можно разными способами. Например, через скалярное произведение векторов можно найти косинусы углов и сравнить их.
![\vec a\cdot \vec b=|\vec a|\cdot |\vec b|\cdot \cos \hat{(\vec a; \vec b)}](https://tex.z-dn.net/?f=%5Cvec+a%5Ccdot+%5Cvec+b%3D%7C%5Cvec+a%7C%5Ccdot+%7C%5Cvec+b%7C%5Ccdot+%5Ccos+%5Chat%7B%28%5Cvec+a%3B+%5Cvec+b%29%7D)
Произвольно построим прямоугольную систему координат XOY. Каждый изображённый угол образуют два вектора. Их координаты легко определить по узлам сетки. Например, координаты вектора ![\vec m=\vec {MA}=(x_M-x_A;y_M-y_A)=(-5-(-4);1-4)=(-1;-3)](https://tex.z-dn.net/?f=%5Cvec+m%3D%5Cvec+%7BMA%7D%3D%28x_M-x_A%3By_M-y_A%29%3D%28-5-%28-4%29%3B1-4%29%3D%28-1%3B-3%29)
1. ∠A; ![\vec a(-1;-1);~~~\vec m(-1; -3)](https://tex.z-dn.net/?f=%5Cvec+a%28-1%3B-1%29%3B~~~%5Cvec+m%28-1%3B+-3%29)
![\boldsymbol{\cos \angle A=}\cos \hat{(\vec a;\vec m)}=\dfrac {-1\cdot (-1)-1\cdot (-3)}{\sqrt{(-1)^2+(-1)^2}\sqrt{(-1)^2+(-3)^2}}=\boldsymbol{\dfrac 2{\sqrt5}}](https://tex.z-dn.net/?f=%5Cboldsymbol%7B%5Ccos+%5Cangle+A%3D%7D%5Ccos+%5Chat%7B%28%5Cvec+a%3B%5Cvec+m%29%7D%3D%5Cdfrac+%7B-1%5Ccdot+%28-1%29-1%5Ccdot+%28-3%29%7D%7B%5Csqrt%7B%28-1%29%5E2%2B%28-1%29%5E2%7D%5Csqrt%7B%28-1%29%5E2%2B%28-3%29%5E2%7D%7D%3D%5Cboldsymbol%7B%5Cdfrac+2%7B%5Csqrt5%7D%7D)
2. ∠B; ![\vec b(1;3);~~~\vec n(1; 1)](https://tex.z-dn.net/?f=%5Cvec+b%281%3B3%29%3B~~~%5Cvec+n%281%3B+1%29)
![\boldsymbol{\cos \angle B=}\cos \hat{(\vec b;\vec n)}=\dfrac {1\cdot 1+3\cdot 1}{\sqrt{1^2+3^2}\sqrt{1^2+1^2}}=\dfrac{4}{\sqrt{10}\sqrt{2}}=\boldsymbol{\dfrac 2{\sqrt5}}](https://tex.z-dn.net/?f=%5Cboldsymbol%7B%5Ccos+%5Cangle+B%3D%7D%5Ccos+%5Chat%7B%28%5Cvec+b%3B%5Cvec+n%29%7D%3D%5Cdfrac+%7B1%5Ccdot+1%2B3%5Ccdot+1%7D%7B%5Csqrt%7B1%5E2%2B3%5E2%7D%5Csqrt%7B1%5E2%2B1%5E2%7D%7D%3D%5Cdfrac%7B4%7D%7B%5Csqrt%7B10%7D%5Csqrt%7B2%7D%7D%3D%5Cboldsymbol%7B%5Cdfrac+2%7B%5Csqrt5%7D%7D)
3. ∠C; ![\vec c(-1;0);~~~\vec p(-3; 1)](https://tex.z-dn.net/?f=%5Cvec+c%28-1%3B0%29%3B~~~%5Cvec+p%28-3%3B+1%29)
![\boldsymbol{\cos \angle C=}\cos \hat{(\vec c;\vec p)}=\dfrac {-1\cdot (-3)+0\cdot 1}{\sqrt{(-1)^2+0^2}\sqrt{(-3)^2+1^2}}=\boldsymbol{\dfrac 3{\sqrt{10}}}](https://tex.z-dn.net/?f=%5Cboldsymbol%7B%5Ccos+%5Cangle+C%3D%7D%5Ccos+%5Chat%7B%28%5Cvec+c%3B%5Cvec+p%29%7D%3D%5Cdfrac+%7B-1%5Ccdot+%28-3%29%2B0%5Ccdot+1%7D%7B%5Csqrt%7B%28-1%29%5E2%2B0%5E2%7D%5Csqrt%7B%28-3%29%5E2%2B1%5E2%7D%7D%3D%5Cboldsymbol%7B%5Cdfrac+3%7B%5Csqrt%7B10%7D%7D%7D)
4. ∠D; ![\vec d(3;-1);~~~\vec q(1; -1)](https://tex.z-dn.net/?f=%5Cvec+d%283%3B-1%29%3B~~~%5Cvec+q%281%3B+-1%29)
![\boldsymbol{\cos \angle D=}\cos \hat{(\vec d;\vec q)}=\dfrac {3\cdot 1-1\cdot (-1)}{\sqrt{3^2+(-1)^2}\sqrt{1^2+(-1)^2}}=\boldsymbol{\dfrac 2{\sqrt5}}](https://tex.z-dn.net/?f=%5Cboldsymbol%7B%5Ccos+%5Cangle+D%3D%7D%5Ccos+%5Chat%7B%28%5Cvec+d%3B%5Cvec+q%29%7D%3D%5Cdfrac+%7B3%5Ccdot+1-1%5Ccdot+%28-1%29%7D%7B%5Csqrt%7B3%5E2%2B%28-1%29%5E2%7D%5Csqrt%7B1%5E2%2B%28-1%29%5E2%7D%7D%3D%5Cboldsymbol%7B%5Cdfrac+2%7B%5Csqrt5%7D%7D)
5. ∠E; ![\vec e(-4;-3);~~~\vec k(-1; -2)](https://tex.z-dn.net/?f=%5Cvec+e%28-4%3B-3%29%3B~~~%5Cvec+k%28-1%3B+-2%29)
![\boldsymbol{\cos \angle E=}\cos \hat{(\vec e;\vec k)}=\dfrac {-4\cdot (-1)-3\cdot (-2)}{\sqrt{(-4)^2+(-3)^2}\sqrt{(-1)^2+(-2)^2}}=\boldsymbol{\dfrac 2{\sqrt5}}](https://tex.z-dn.net/?f=%5Cboldsymbol%7B%5Ccos+%5Cangle+E%3D%7D%5Ccos+%5Chat%7B%28%5Cvec+e%3B%5Cvec+k%29%7D%3D%5Cdfrac+%7B-4%5Ccdot+%28-1%29-3%5Ccdot+%28-2%29%7D%7B%5Csqrt%7B%28-4%29%5E2%2B%28-3%29%5E2%7D%5Csqrt%7B%28-1%29%5E2%2B%28-2%29%5E2%7D%7D%3D%5Cboldsymbol%7B%5Cdfrac+2%7B%5Csqrt5%7D%7D)
cos ∠A = cos ∠B = cos ∠D = cos ∠E =![\boldsymbol{\dfrac 2{\sqrt5}}](https://tex.z-dn.net/?f=%5Cboldsymbol%7B%5Cdfrac+2%7B%5Csqrt5%7D%7D)
Ответ: ∠A = ∠B = ∠D = ∠E