a Use the fact that the lines intersect outside the circle to write the corresponding equation. Use the Arc Addition Postulate. Notice that mJH must be greater than 90^(∘) and less than or equal to 180^(∘).
B
b Use the Arc Addition Postulate and the fact that GJ and GH intersect each other outside the circle.
A
aRange of Values: m∠ G ≤ 90^(∘) Explanation: See solution.
B
bArcs Measures: mHJ=124^(∘) and mKH=56^(∘) Explanation: See solution.
Practice makes perfect
a We are given the figure below, where JK is a diameter and GH is a tangent to the circle.
From the above we have that the measure of ∠ G is equal to half the measure of the difference of the intercepted arcs.
m∠ G = 1/2(m JH - m HK)Now, since JK is a diameter we have that mJHK = 180^(∘). Using this and the Arc Addition Postulate, we get the following equation.
m JH + m HK = mJHK^(180^(∘))
⇓
m HK = 180^(∘) - m JH
Let's substitute this equation into the first one written above.
Next, to describe the range of possible values for m∠ G we need to study the possible values for m JH.
Notice that as m JH becomes closer to 90^(∘), we have that GJ becomes almost parallel to GH, and in this case the lines will not intersect each other. On the other hand, the maximum value for m JH is 180^(∘).
90^(∘) < m JH ≤ 180^(∘)
Next, let's subtract 90^(∘) in each side of the inequality above.
0^(∘) < m JH-90^(∘) ≤ 90^(∘)
⇓
0 < m∠ G ≤ 90^(∘)
Thus, m∠ G can be any angle with a measure greater than 0^(∘) and less than or equal to 90^(∘). We have that m∠ G=90^(∘) when m JH=180^(∘) — that is, when GJ⊥GH.
b Since GJ and GH intersect each other outside the circle and they are secant and tangent to it, we get the following equation.
Also, because JK is a diameter of the circle we have that mJHK=180^(∘). Using this and the Arc Addition Postulate, we can write the following equation.
m JH + m HK = mJHK^(180^(∘))
⇓
m HK = 180^(∘) - m JH
Let's substitute this equation and m∠ G=34^(∘) into the one written at the beginning to find the measure of m JH.
