Question

1. in the diagram, which of the following pairs does not represent the same geometric object? (1) \\( \overline { c g } \\) and \\( \overline { g c } \\) (2) \\( \overline { f c } \\) and \\( \overline { f d } \\) (3) \\( \overline { e a } \\) and \\( \overline { a e } \\) (4) \\( \angle b a g \\) and \\( \angle f a b \\) 3. points a, e, and d are collinear and rays \\( \overline { e b } \\) and \\( \overline { e c } \\) are perpendicular, as shown in the diagram below. if \\( m \angle c e d = 17 ^ { \circ } \\), which of the following is the measure of \\( \angle b e a \\)? (1) \\( 17 ^ { \circ } \\) (2) \\( 73 ^ { \circ } \\) (3) \\( 90 ^ { \circ } \\) (4) \\( 153 ^ { \circ } \\) 4. it is known that \\( m \angle j = 4 x - 17 \\) and \\( m \angle k = 2 x + 11 \\). if \\( \angle j \\) and \\( \angle k \\) are complementary angles, then which of the following is the value of x? (1) 14 (2) 16 (3) 29 (4) 31

Explanation:

Step1: Analyze each option for question 2

• Option (1):

A line segment \(\overline{CG}\) has endpoints \(C\) and \(G\). The line segment \(\overline{GC}\) also has endpoints \(G\) and \(C\). Since the order of endpoints does not matter for a line segment, \(\overline{CG}\) and \(\overline{GC}\) represent the same geometric object.

• Option (2):

A ray \(\overline{FC}\) has endpoint \(F\) and passes through \(C\), while a ray \(\overline{FD}\) has endpoint \(F\) and passes through \(D\). Since they have the same endpoint \(F\) but pass through different points \(C\) and \(D\), they are different rays.

• Option (3):

A line \(\overline{EA}\) and \(\overline{AE}\) represent the same line. A line has no direction, and the notation \(\overline{EA}\) and \(\overline{AE}\) just name two points on the same line.

• Option (4):

\(\angle BAG\) and \(\angle FAB\) are the same angle. They have the same vertex \(A\) and the same two - side rays (the rays forming the angle).

Step2: Analyze question 3

Since \(A\), \(E\), and \(D\) are collinear, \(\angle AEC+\angle CED = 180^{\circ}\) (linear - pair of angles). Given \(\angle CED=17^{\circ}\), then \(\angle AEC = 180^{\circ}-\angle CED=163^{\circ}\). Also, since \(\overline{EB}\perp\overline{EC}\), \(\angle BEC = 90^{\circ}\). Then \(\angle BEA=\angle AEC-\angle BEC\). Substitute the values: \(\angle BEA=163^{\circ}- 90^{\circ}=73^{\circ}\)

Step3: Analyze question 4

If \(\angle J\) and \(\angle K\) are complementary angles, then \(m\angle J + m\angle K=90^{\circ}\). Substitute \(m\angle J = 4x - 17\) and \(m\angle K=2x + 11\) into the equation: \((4x-17)+(2x + 11)=90\). Combine like terms: \(4x+2x-17 + 11=90\), \(6x-6 = 90\). Add \(6\) to both sides: \(6x=90 + 6=96\). Divide both sides by \(6\): \(x=\frac{96}{6}=16\)

Answer:

For question 2: (2) \(\overline{FC}\) and \(\overline{FD}\)
For question 3: (2) \(73^{\circ}\)
For question 4: (2) \(16\)