Geometry Question

Question

In the diagram, Triangle ZAB is inscribed in Circle O such that segment ZB is a diameter of the circle.

Abigail writes a two column proof to show that angle ZAB is a right angle, but she is missing a few steps.

Read her proof carefully, and then answer the question.

1) Triangle ZAB is inscribed in Circle O; segment ZB is a diameter. 1) Given

2) [Blank 1] 2) All radii of the same circle are congruent.

3) Triangle ZOA and BOA are isosceles triangles. 3) Definition of Isosceles Triangle

4) Since angle AZO is congruent to angle OAZ, let x degrees= angle measure of AZO and x degrees= angle measure of OAZ. Since angle OBA Is congruent to angle OAB, let y degrees= angle measure OBA and y degrees = angle measure of OAB. 4) [Blank 2]

5) Angle measure of AOB= x degrees+x degrees=(2x) degrees; angle measure of ZOA= y degrees+y degrees=(2y) degrees 5) [Blank 3] (and simplify)

6) Angle ZOA and Angle BOA are a linear pair and supplementary. 6) Definition of linear pair.

7) Angle Measure of ZOA+ Angle Measure of BOA= 180 degrees 7) Definition of supplementary

8) (2x) degrees + (2y) degrees = 180 degrees 8) Substitution Property of Equality

9) x degrees + y degrees = 90 degrees 9) Division Property of Equality.

10) Angle Measure of ZAB= Angle Measure of ZAO + Angle Measure of OAB 10) [Blank 4]

11) Angle Measure of ZAB= x degrees + y degrees; and measure of ZAB= 90 degrees 11) Substitution Property of Equality

12) Angle ZAB is a right angle. 12) Definition of right angle

Which statements and reasons complete her proof most logically?

blank 1: segmentOZ is congruent to segment OA which is congruent to segment OB

blank 1: segmentOZ is congruent to segment AB which is congruent to segment OB

blank 2: Triangle Exterior Angles Theorem

blank 2: Base Angles Theorem

blank 3: Triangle Exterior Angle Theorem

blank 3: Base Angles Theorem

blank 4: Triangle Sum Theorem

blank 4: Angle Addition Postulate

Answer 1

As long as angles α(alpha) and β(beta) =90°; you've got a right triangle. γ(gamma) is the rt < and is positioned away from the diameter.