Question

elena and diego are working together on this problem: here is a figure where ray r meets line l. the dashed rays are angle bisectors. diego makes this conjecture: \the angle formed between the angle bisectors is always a right angle, no matter what the angle between r and l is.\ elena says, \its difficult to tell specifically which angles youre talking about.\ she labels the diagram and restates the conjecture as the following:
ay ce bisects angle acd into two congruent angles. ray cf bisects angle bcd into two congruent angles. we conjecture angle ecf is a right angle.\ diego adds more information to the diagram as he tells elena, \we can put letters here to represent the angle measures. so these two angles are each a°, and these are b°. that means our conjecture is a + b = 90.\ elena exclaims, \oh! i see it now. angle acb measures 180°, so a + a + b + b = 180. then the middle part has to be a right angle.\ diego writes down a summary of their conversation: \for any straight line l and ray r, the angle bisectors of the two angles form a right angle. thats because there are two pairs of congruent angles, a° and b°, that sum to 180°. so a° + b° has to equal 90°, a right angle.\

Explanation:

Step1: Analyze the angle sum

Since \(ACB\) is a straight - line angle, \(\angle ACB = 180^{\circ}\). Given that \(CE\) bisects \(\angle ACD\) (so \(\angle ACE=\angle ECD = a^{\circ}\)) and \(CF\) bisects \(\angle BCD\) (so \(\angle DCF=\angle FCB = b^{\circ}\)), we have \(\angle ACE+\angle ECD+\angle DCF+\angle FCB=180^{\circ}\), which is \(2a + 2b=180^{\circ}\).

Step2: Solve for \(a + b\)

Divide the equation \(2a + 2b = 180^{\circ}\) by \(2\). Using the formula \(\frac{2a+2b}{2}=\frac{180^{\circ}}{2}\), we get \(a + b=90^{\circ}\). And \(\angle ECF=\angle ECD+\angle DCF=a + b\).

Answer:

The angle formed between the angle bisectors is a right angle.