Question

На сторонах АВ, ВС, АС треугольника АВС отмечены точки Т, Р, М соответственно угол МРС= 51 градусу, угол АВС=52 градуса, угол АТМ=52 градуса. Найдите угол ТМР? Докажите, что прямые МР И ВС имеют общую точку.

Answer 1

1) Тр-ки АВС и АТМ подобны (по стороне и 2-ум прилегающим к ней углам) , след-но 
ТМ // ВС . При секущей МР углы МРС и ТМР равны, как внутр-ние накрест лежащие, значит угол ТМР=51град. 
2) угол MPC=51 градус, угол ABC=52 градуса, значит МР и ВТ не парал-ны, след-но пересекутся.

Answer 2

Given Information:

- Triangle ABC with points T, R, and M marked on sides AB, BC, and AC respectively. - Angle MRS = 51 degrees. - Angle ABC = 52 degrees. - Angle ATM = 52 degrees.

Finding Angle TMR:

To find angle TMR, we can use the fact that the sum of the angles in a triangle is 180 degrees. Since we know the measures of angles MRS and ATM, we can find angle TMR by subtracting the sum of these two angles from 180 degrees.

Let's calculate the measure of angle TMR:

Angle TMR = 180 degrees - (Angle MRS + Angle ATM) = 180 degrees - (51 degrees + 52 degrees) = 180 degrees - 103 degrees = 77 degrees.

Therefore, angle TMR is 77 degrees.

Proving that lines MR and BC have a common point:

To prove that lines MR and BC have a common point, we can use the fact that if two lines intersect, then the opposite angles formed are equal.

Let's consider the angles formed by lines MR and BC:

Angle MRS (given) = 51 degrees. Angle ABC (given) = 52 degrees.

If lines MR and BC intersect at point P, then we can say that:

Angle MRP = Angle ABC (opposite angles formed by intersecting lines)

Since Angle MRP = 51 degrees and Angle ABC = 52 degrees, we can conclude that lines MR and BC intersect at point P.

Therefore, lines MR and BC have a common point.