Question

a student uses the vertical angles theorem and the definition of complementary angles to conclude ( mangle ptr = 50 ) in the figure. what mistake did the student make?

choose the correct answer below

a. the student used the definition of complementary angles instead of the definition of right angles.

b. the student used the definition of complementary angles instead of the definition of vertical angles

c. the student used the definition of complementary angles instead of the definition of supplementary angles

d. the student used the definition of vertical angles instead of the definition of right angles

Explanation:

Vertical angles are equal. But here, \((x + 28)^{\circ}\) and \((2x+18)^{\circ}\) are vertical angles. Complementary angles add up to \(90^{\circ}\), while supplementary angles add up to \(180^{\circ}\). Since these are vertical angles (which are equal) and they form a linear - pair (sum to \(180^{\circ}\) when adjacent, but here as vertical angles, we first find \(x\) using the equality of vertical angles \(x + 28=2x + 18\) (wrong approach if using complementary). The correct relation for vertical angles (they are equal) and for a linear - pair (sum to \(180^{\circ}\) when adjacent, but here vertical angles \(\angle QTP=(x + 28)^{\circ}\) and \(\angle STR=(2x + 18)^{\circ}\), and \(\angle QTP+\angle PTR = 180^{\circ}\) (supplementary, linear - pair). The student should have used supplementary (sum to \(180^{\circ}\)) instead of complementary (sum to \(90^{\circ}\)).

Answer:

B. The student used the definition of complementary angles instead of the definition of supplementary angles