We will prove Theorem 5-10 by filling in the blanks for the given two-column proof. Let's first recall what the theorem says.
If two sides of a triangle are not congruent,
then the larger angle lies opposite the longer side.
Now, let's examine the given information!
Given: △ TOY, with YO>YT
What does the given information indicate? This is the mathematical expression of the hypothesis of the theorem. In this case, we need to prove the conclusion of the theorem.
Blanks a. and b.
The first two missing pieces of information need to be completed by a statement that will eventually be our conclusion. In this case, we want to prove that the angle that lies opposite YO is larger than the angle that lies opposite YT.
Prove: a. m∠ OTY> b. m∠ 3
Typically in a two-column proof, the Prove statement is the same as the final statement in the proof.
Blank c.
As we can see by the markings, △ TPY is an isosceles triangle. Thus, we need to use the Isosceles Triangle Theorem to prove the second step.
2)& m∠1=m∠2
2)& c. Isosceles Triangle Theorem
Blanks d. and e.
In the figure, point T is in the interior of ∠ OTY. We can write m∠ OTY as the sum of m∠ OTP and m∠ YTP. Thus, we can prove the third step by the Angle Addition Postulate.
3)& m∠ OTY=m∠4+m∠2
3)& d. Angle Addition Postulate
Since the fourth step is the comparison of the angles from the third step, the Comparison Property of Inequality proves this step.
4)& m∠ OTY>m∠2
4)& e. Comparison Property of Inequality
Blank f.
In the second step, we proved that m∠1=m∠2. Therefore, by substituting m∠ 1 for m∠ 2 into the fourth step, we can prove the fifth step.
5)& m∠ OTY>m∠1
5)& f. Substitution Property of Equality
Blank g.
For the next step, let's examine the figure once more.
In the figure, we see that ∠ 1 is the exterior angle of △ TOP and ∠ 3 and ∠ 4 are remote interior angles of ∠ 1. Thus, the Corollary to the Triangle Exterior Angle Theorem is what we need!
6)& m∠ 1>m∠3
6)& g. Corollary to the Triangle
& Exterior Angle Theorem
Blank h.
The final step of the proof is the result of steps five and six. In order to combine these steps, we will use the Transitive Property of Inequality.
7)& m∠ OTY>m∠3
7)& h. Transitive Property of Inequality
Completed Proof
Let's write all the information together to create our two-column proof!
