Question

multiple - choice questions

1. in the diagram above, four straight lines intersect at point x. ( a^{circ}=b^{circ}=60^{circ} ). what is ( c^{circ}+d^{circ} )?

a. ( 30^{circ} )
b. ( 45^{circ} )
c. ( 55^{circ} )
d. ( 60^{circ} )

1. in the diagram above, line 1 intersects parallel lines 2 and 3 and parallel lines 4 and 5, such that ( a^{circ} ) is equal to ( b^{circ} ). what is the value of ( c )?

e. ( 180^{circ}-a^{circ} )
f. ( 180^{circ}-b^{circ} )
g. ( 180^{circ}-a^{circ}-b^{circ} )
h. ( a^{circ} )

1. in the diagram above, ( a b ) is congruent to ( b c ), and ( x ) is the midpoint of ( a c ). if ( a b = 4 \frac{3}{8} ) inches and ( a x = 3 \frac{1}{4} ) inches, what is the perimeter of triangle ( a b c )?

a. ( 7 \frac{5}{8} ) in.
b. ( 10 \frac{7}{8} ) in.
c. ( 12 ) in.
d. ( 15 \frac{1}{4} ) in.

1. the length of a rectangle is ( 4 x - 7 ), and its width is 6. if the area of the rectangle is 54, what is the value of ( x )?

e. 3
f. 4
g. 5
h. 7

Explanation:

Step1: Find the measure of \(c + d\)

Since \(a = b=60^{\circ}\), and the sum of angles around a point is \(360^{\circ}\). Also, vertical angles are equal. The sum of all angles around point \(X\) is \(360^{\circ}\). We know that \(a + b+(c + d)+ 60^{\circ}+60^{\circ}=360^{\circ}\) (because of vertical - angle relationships).
Substitute \(a = 60^{\circ}\) and \(b = 60^{\circ}\) into the equation: \(60^{\circ}+60^{\circ}+(c + d)+60^{\circ}+60^{\circ}=360^{\circ}\).
Simplify the left - hand side: \(240^{\circ}+(c + d)=360^{\circ}\).

Step2: Solve for \(c + d\)

Subtract \(240^{\circ}\) from both sides of the equation: \(c + d=360^{\circ}-240^{\circ}\).

Answer:

\(D.60^{\circ}\)