For each case, we can write a Two-column proof lists each statement on the left and the justification is on the right. Each statement must follow logically from the steps before it. Let's start with the first case!
Given that a tangent and a secant intersect outside a circle, we will show that the measure of ∠ 1 is half the difference of the measures of BC and AC. In a two-column proof we always start by stating the given information.
Statement1)& A tangent and a secant
& intersect outside a circle
Reason1)& Given
We will continue by writing the measure of angle ∠ 1 in terms of the measures of ∠ ABC and ∠ 2. Therefore, we can immediately decide the next steps of the proof. Notice that ∠ 2 is an of the triangle with two nonadjacent angles, ∠ 1 and ∠ ABC.
By the Exterior Angle Theorem (Theorem 5.2), the measure of ∠ 2 is the sum of the measures of ∠ 1 and ∠ ABC.
Statement2)& m∠ 2 = m∠ 1 + m∠ ABC
Reason2)& Exterior Angle Theorem
Now, by the we can isolate m∠ 1.
Statement3)& m∠ 1 = m∠ 2 - m∠ ABC
Reason3)& Subtraction Property
& of Equality
Next, we will write m∠ 2 and m∠ ABC in terms of their intercepted arcs. Looking at the diagram, we can see that ∠ 2 is formed by the intersection of a tangent and a , and it intercepted arc is BC.
In this case we should consider the (Theorem 10.14).
|
Tangent and Intersected Chord Theorem
|
|
If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one-half the measure of its intercepted arc.
|
By this theorem, the measure of ∠ 2 is half the measure of BC.
Statement4)& m∠ 2 = 1/2mBC
Reason4)& Tangent and Intersected
& Chord Theorem
Now we will investigate ∠ ABC. As we can see, ∠ ABC is an whose intercepted arc is AC.
By the (Theorem 10.10), we can deduce that the measure of ∠ ABC is half the measure of AC.
Statement5)& m∠ ABC = 1/2mAC
Reason5)& Inscribed Angle Theorem
From here we can substitute m∠ 2 = 12mBC and m∠ ABC = 12mAC into the third statement by the .
Statement6
m∠ 1 = 1/2mBC - 1/2mAC [1.1em]
Reason6
Substitution Property of Equality
Finally, by the we will factor out 12 and complete our proof.
Statement7)& m∠ 1 = 1/2(mBC - mAC)
Reason7)& Distributive Property
Let's summarize the above process in a two-column table.
|
|
|
|
|
|
|
3.Subtraction Property of Equality
|
|
4.Tangent and Intersected Chord Theorem
|
|
5.Inscribed Angle Theorem
|
|
6.Substitution Property of Equality
|
|
7.Distributive Property of Equality
|
Case II
In this case, we will investigate the measure of an angle formed by two tangents intersecting outside the circle.
This time, both ∠ 2 and ∠ 3 are the angles formed by the intersection of a tangent line and a chord. Additionally, the intercepted arcs of ∠ 2 and ∠ 3 are PR and PQR.
Keeping this information in mind and proceeding in the same way as we proved Case I, we can write a two-column proof for Case II as follows.
|
|
|
|
|
|
|
3.Subtraction Property of Equality
|
|
4.Tangent and Intersected Chord Theorem
|
|
5.Tangent and Intersected Chord Theorem
|
|
6.Substitution Property of Equality
|
|
7.Distributive Property of Equality
|
Case III
Finally, we will prove the last case. Let's draw a diagram that models it!
In this case, ∠ XYT and ∠ 2 are inscribed angles whose intercepted arcs are XT and YZ.
Again, we will use the same steps as we did in the previous cases to write a two-column proof for Case III.
|
|
|
|
|
|
|
3.Subtraction Property of Equality
|
|
4.Inscribed Angle Theorem
|
|
5.Inscribed Angle Theorem
|
|
6.Substitution Property of Equality
|
|
7.Distributive Property of Equality
|
