Question

in triangle cde, the measure of angle c is ( x^circ ). the measure of angle d is ( 40^circ ). which of the following expressions could be used to represent the value of angle e?

reference sheet

diagrams: circle (radius ( r )), rectangle (( l, w )), triangle (( b, h )), right triangle (( a, b, c )), 30-60-90 triangle, 45-45-90 triangle, rectangular prism (( l, w, h )), cylinder (( r, h )), sphere (( r )), cone (( r, h )), pyramid (( l, w, h )) with corresponding formulas: ( a = pi r^2, c = 2pi r ); ( a = lw ); ( a = \frac{1}{2}bh ); ( c^2 = a^2 + b^2 ); special right triangle ratios; ( v = lwh ); ( v = pi r^2 h ); ( v = \frac{4}{3}pi r^3 ); ( v = \frac{1}{3}pi r^2 h ); ( v = \frac{1}{3}lwh ). text: the number of degrees of arc in a circle is 360. the number of radians of arc in a circle is ( 2pi ). the sum of the measures in degrees of the angles of a triangle is 180.

options:
( (x - 40 - 180)^circ )
( (180 - 40 + x)^circ )
( 180 - (40 + x)^circ )
( 180 + (40 + x)^circ )

Explanation:

Step1: Recall triangle angle sum

The sum of angles in a triangle is \(180^\circ\). So, \(\angle C + \angle D + \angle E = 180^\circ\).

Step2: Substitute known values

Given \(\angle C = x^\circ\), \(\angle D = 40^\circ\). Substitute into the formula: \(x + 40 + \angle E = 180\).

Step3: Solve for \(\angle E\)

Subtract \(x\) and \(40\) from both sides: \(\angle E = 180 - (x + 40)\), so \(\angle E = [180 - (40 + x)]^\circ\).

Answer:

\([180 - (40 + x)]^\circ\) (the third option)