Chapter 10 - Geometry - 10.2 Triangles - Exercise Set 10.2 - Page 626: 6

Measurement of angle 1 is\[40{}^\circ \], angle 2 is\[140{}^\circ \], angle 3 is\[40{}^\circ \], angle 4 is\[140{}^\circ \], and angle 5 is\[35{}^\circ \].

Suppose that the angle thatis supplementary to the right angle is\[\angle 6\]. Since, \[\angle 6\]is supplementary to the right angle thus\[m\angle 6=90{}^\circ \]. According to angle sum property, the sum of all the three angles of a triangle is\[180{}^\circ \]. Compute the measure of angle 1 as follows: Sum of all the three angles of a triangle is\[180{}^\circ \]. \[m\angle 1+m\angle 6+50{}^\circ =180{}^\circ \] Simplify the equation by substituting the given values as follows: \[\begin{align} & m\angle 1+90{}^\circ +50{}^\circ =180{}^\circ \\ & m\angle 1+140{}^\circ =180{}^\circ \\ \end{align}\] Compute the measurement of angle 1 by subtracting \[140{}^\circ \]from both the sides of equation as shown below: \[\begin{align} & m\angle 1=180{}^\circ -140{}^\circ \\ & m\angle 1=40{}^\circ \\ \end{align}\] Compute the measurement of angle 3 using the fact that vertically opposite angles are equal to each other as shown below: \[\begin{align} & m\angle 3=m\angle 1 \\ & m\angle 3=40{}^\circ \\ \end{align}\] According to angle sum property, the sum of all the three angles of a triangle is\[180{}^\circ \]. Compute the measure of angle 5 as follows: Sum of all the three angles of a triangle is\[180{}^\circ \]. \[\angle 3+\angle 5+105{}^\circ =180{}^\circ \] Simplify the equation by substituting the given values as follows: \[\begin{align} & 40+m\angle 5+105{}^\circ =180{}^\circ \\ & m\angle 5+145{}^\circ =180{}^\circ \end{align}\] Compute the measurement of angle 5 by subtracting \[145{}^\circ \]from both the sides of equation as shown below: \[\begin{align} & m\angle 5=180{}^\circ -145{}^\circ \\ & m\angle 5=35{}^\circ \\ \end{align}\] Compute the measurement of angle 2 using the fact that angles 3 and 2 form a straight angle. Therefore, their sum is equal to\[180{}^\circ \]. \[m\angle 3+m\angle 2=180{}^\circ \] Simplify the equation by substituting the given values as follows: \[40{}^\circ +m\angle 2=180{}^\circ \] Compute the measurement of angle 2 by subtracting \[{{40}^{{}^\circ }}\]from both the sides of equation. \[\begin{align} & m\angle 2=180{}^\circ -40{}^\circ \\ & \angle m2=140{}^\circ \\ \end{align}\] Compute the measurement of angle 4 using the fact that vertically opposite angles are equal to each other as shown below. \[\begin{align} & m\angle 4=m\angle 2 \\ & m\angle 4=140{}^\circ \\ \end{align}\]