Begin by drawing a diameter of ⊙ E that intersects l at H.
See solution.
Practice makes perfect
Given that GH and tangent l are intersecting ⊙ E at H, we will prove that m∠ GHI= 12mGFH. To do so, we will write a two-column proof.
Two-column proofs list each statement on the left and the justification on the right. Each statement must follow logically from the steps before it. In a two-column proof, we always start by stating the given information.
Statement1)& GH and tangent l
& intersecting ⊙ E atH.
Reason1)& Given
Our next step will be drawing a diameter of ⊙ E that intersects l at H.
Thus, JH is a diameter of ⊙ E by the definition of a diameter.
Statement2)& JH is a diameter
Reason2)& Definition of a Diameter
Now, by Theorem 12-1 we can conclude that JH is perpendicular to HI.
Statement3)& JH ⊥ HI
Reason3)& Theorem12-1
By the definition of perpendicular lines, ∠ JHI is right angle.
Statement4)& ∠ JHI is a right angle.
Reason4)& Definition of Perpendicular Lines
Consequently, the measure of ∠ JHI is 90^(∘) by the definition of a right angle.
Statement5)& m∠ JHI= 90^(∘)
Reason5)& Definition of a Right Angle
The purpose of the above steps was to find the measure of ∠ JHI.
From here, by the Angle Addition Postulate we can write m∠ JHI as the sum of m∠ JHG and m∠ GHI.
Statement6)& m∠ JHG + m∠ GHI= 90^(∘)
Reason6)& Angle Addition Postulate
Notice that m∠ JHG is an inscribed angle and its intercepted angle is mJG.
Therefore, be the Inscribed Angle Theorem the measure of ∠ JHG is half the measure of JG.
Statement7)& m∠ JHG = 1/2mJG
Reason7)& Inscribed Angle Theorem
Now that we found m∠ JHG in terms of 12mJG, we can substitute it into m∠ JHG + m∠ GHI= 90^(∘) using the Substitution Property of Equality.
Statement8)& 12mJG + m∠ GHI= 90^(∘)
Reason8)& Substitution Property
&of Equality
Next, we will write mJG in terms of mGFH. Notice that JG and GFH form a semicircle.
Therefore, by the Arc Addition Postulate the sum of JG and GFH is 180^(∘).
Statement9)& JG+GFH=180^(∘)
Reason9)& Arc Addition Postulate
Let's isolate mJG by subtracting mGFH from both sides of the equation by the Subtraction Property of Equality.
Statement10)& JG=180^(∘)-GFH
Reason10)& Subtraction Property
&of Equality
Then, we will substitute it into 12mJG+ m∠ GHI= 90^(∘).
Statement11
1/2(180^(∘)-GFH) + m∠ GHI= 90^(∘) [1.1em]
Reason11
Substitution Property of Equality
In the last part of our proof, we will isolate m∠ GHI by using the Properties of Equality. Let's first distribute 12 by the Distributive Property.
Statement12
90^(∘)-1/2GFH + m∠ GHI= 90^(∘) [1.1em]
Reason12
Distributive Property
By the Subtraction Property of Equality, we will subtract 90 ^(∘) from both sides of the equation.
Statement13)& -1/2GFH + m∠ GHI= 0
Reason13)& Subtraction Property
&of Equality
Finally, we will add 12mGFH to both sides of the equation and complete our proof by the Addition Property of Equality.
Statement14)& m∠ GHI= 1/2GFH
Reason14)& Addition Property
&of Equality
Let's summarize the above process in a two-column table.
