Range of Possible Angle Measures: m∠ LPJ < 90^(∘) Explanation: See solution.
Practice makes perfect
Let's recall the given diagram!
We have been told that PL is tangent to the circle and KJ is a diameter. We want to find the range of possible angle measures of ∠ LPJ. Notice that we have a tangent and a secant intersecting outside the circle. Thus, we will first recall the Angles Outside the Circle Theorem (Theorem 10.16).
Angles Outside the Circle Theorem
If a tangent and a secant, two tangents, or two secants intersect outside a circle, then the measure of the angle formed is one-half the difference of the measures of the intercepted arcs.
Using this theorem, we can express the measure of ∠ LPJ in terms of mLK and mLJ as follows.
m∠ LPJ = 1/2(mLJ-mLK)
Next, we will find the range of possible measures of LK and LJ. Since KJ is a diameter, we can immediately conclude that by the definition of a semicircle, KLJ is a semicircle.
The measure of a whole circle is 360^(∘), so the measure of a semicircle is 180^(∘). Therefore, by the Arc Addition Postulate (Postulate 10.1), the sum of mLK and mLJ is 180^(∘).
With this information we can conclude that both mLK and mLJ are less than 180^(∘). Consequently, the difference of them is also less than 180^(∘).
lmLK< 180^(∘) mLJ<180^(∘) ⇒ mLK-mLJ < 180^(∘)
Now that we know the range of the possible measures of mLK-mLJ, we can find the range of possible angle measures of 12(mLJ-mLK).
mLK-mLJ < 180^(∘)
⇕
1/2(mLJ-mLK) < 90^(∘)
Finally, by substituting m∠ LPJ for 12(mLJ-mLK), we can find the range of possible angle measures of ∠ LPJ.
m∠ LPJ < 90^(∘)
