7 knacc Упражнения по матоматико (7) В треугольнике \( A B C, \angle C=85^{\circ} \) и \( \angle A: \angle B=2: 3 \). Найдите разность 4. В треугольнике \( A B C, \angle C=\angle A+50^{\circ} \) и \( \angle A: \angle B=3: 4 \). Найдите градусную меру угла \( \angle B \). (5.) В треугольнике \( A B C, \angle A: \angle B: \angle C=3: 4: 5 \). Найдите разность \( \angle \mathrm{C}-\angle B \). 6. В треугольнике \( A B C, \angle A=\angle B-21^{\circ}, \angle B=\angle C-18^{\circ} \). Найдите градусную меру угла \( C \). 7. В треугольнике \( A B C, \angle B=\angle A+29^{\circ} \) и \( \angle \mathrm{C}=\frac{3}{4} \angle B \). Найдите градусную меру угла \( C \).

Alright, I need to solve these geometry problems involving triangles and angle measures. Let's tackle them one by one. **Problem 4:** In triangle \( ABC \), angle \( C = 85^\circ \) and the ratio of angles \( A : B = 2 : 3 \). Find the difference \( \angle C - \angle A \). First, I recall that the sum of angles in a triangle is \( 180^\circ \). So, \( \angle A + \angle B + \angle C = 180^\circ \). Given: - \( \angle C = 85^\circ \) - \( \angle A : \angle B = 2 : 3 \) Let's denote \( \angle A = 2x \) and \( \angle B = 3x \). So, \( 2x + 3x + 85^\circ = 180^\circ \) \( 5x + 85^\circ = 180^\circ \) \( 5x = 180^\circ - 85^\circ \) \( 5x = 95^\circ \) \( x = 19^\circ \) Therefore: - \( \angle A = 2x = 38^\circ \) - \( \angle B = 3x = 57^\circ \) Now, the difference \( \angle C - \angle A = 85^\circ - 38^\circ = 47^\circ \). **Problem 5:** In triangle \( ABC \), the ratio of angles \( A : B : C = 3 : 4 : 5 \). Find the difference \( \angle C - \angle B \). Again, the sum of angles in a triangle is \( 180^\circ \). Given: - \( \angle A : \angle B : \angle C = 3 : 4 : 5 \) Let's denote: - \( \angle A = 3x \) - \( \angle B = 4x \) - \( \angle C = 5x \) So, \( 3x + 4x + 5x = 180^\circ \) \( 12x = 180^\circ \) \( x = 15^\circ \) Therefore: - \( \angle A = 45^\circ \) - \( \angle B = 60^\circ \) - \( \angle C = 75^\circ \) The difference \( \angle C - \angle B = 75^\circ - 60^\circ = 15^\circ \). **Problem 6:** In triangle \( ABC \), \( \angle A = \angle B - 21^\circ \) and \( \angle B = \angle C - 18^\circ \). Find the measure of angle \( C \). Let's denote: - \( \angle A = \angle B - 21^\circ \) - \( \angle B = \angle C - 18^\circ \) From the second equation, \( \angle C = \angle B + 18^\circ \) Substitute \( \angle B \) from the second equation into the first: \( \angle A = (\angle C - 18^\circ) - 21^\circ = \angle C - 39^\circ \) Now, sum of angles: \( \angle A + \angle B + \angle C = 180^\circ \) \( (\angle C - 39^\circ) + (\angle C - 18^\circ) + \angle C = 180^\circ \) \( 3\angle C - 57^\circ = 180^\circ \) \( 3\angle C = 237^\circ \) \( \angle C = 79^\circ \) **Problem 7:** In triangle \( ABC \), \( \angle B = \angle A + 29^\circ \) and \( \angle C = \frac{3}{4} \angle B \). Find the measure of angle \( C \). Given: - \( \angle B = \angle A + 29^\circ \) - \( \angle C = \frac{3}{4} \angle B \) Sum of angles: \( \angle A + \angle B + \angle C = 180^\circ \) Substitute \( \angle B \) and \( \angle C \): \( \angle A + (\angle A + 29^\circ) + \frac{3}{4}(\angle A + 29^\circ) = 180^\circ \) Let's denote \( \angle A = x \): \( x + (x + 29^\circ) + \frac{3}{4}(x + 29^\circ) = 180^\circ \) \( x + x + 29^\circ + \frac{3}{4}x + \frac{87^\circ}{4} = 180^\circ \) Combine like terms: \( 2x + \frac{3}{4}x + 29^\circ + 21.75^\circ = 180^\circ \) \( \frac{11}{4}x + 50.75^\circ = 180^\circ \) \( \frac{11}{4}x = 129.25^\circ \) \( x = \frac{129.25^\circ \times 4}{11} \) \( x = \frac{517^\circ}{11} \) \( x \approx 47.0^\circ \) Now, \( \angle B = x + 29^\circ = 47^\circ + 29^\circ = 76^\circ \) \( \angle C = \frac{3}{4} \times 76^\circ = 57^\circ \) So, \( \angle C = 57^\circ \).