You may use pencil, pens, protractor, ruler, compass and scissors on this test. You may not use a calculator, book or notes. Please take your time on each problem and show your work in the space provided. If you get stuck on a problem, move on to the next and return when you have time.
Answer: Use the Star Trek Lemma. Angles RQ'Q and QR'R are inscribed angles. They span the same arc: RQ. If O is the center of the circle, the Star Trek Lemma says that the measure of angle RQ'Q is half that of angle ROQ and also says that the measure of angle QR'R is half that of angle ROQ. Therefore angle RQ'Q is congruent to angle QR'R.
Claim: If D is the midpoint of segment AC, then triangles ADB and A'DC are congruent.
Proof: We use Side-Side-Angle to prove congruence.
Angles ADB and A'DC are corresponding angles; therefore they are congruent.
By definition of midpoint, Segment A'D and segment DB are congruent.
Because angles ADB and A'DC are congruent, the Star Trek Lemma tells us that arc AB is congruent to arc A'C. Therefore segment (chord) AB is congruent to segment A'C.
By Side-Side-Angle, triangles ADB and A'DC are congruent.//
You cannot prove congruence using Side-Side-Angle except in special cases.
Angles ADB and A'DC are vertical angles, not corresponding angles.
D is the midpoint of AC, so AD is congruent to DC. We do not know enough to compare A'D and DB.
The Star Trek Lemma only discusses angles whose vertices are on the circle or on the center of the circle. It does not apply here.
Answer: The proof is given on pages 34-35 of the text.
Answer: The construction is part of the proof of lemma 3.3.1 in the text.