Exercise 19 Page 430

Consider Theorem 5.10 about Angle-Side Relationships in Triangles.

Angles: ∠ C, ∠ D, ∠ E
Sides: DE, CE, CD

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To find the order of angle measures and sides from smallest to largest, let's look at the given diagram and consider the given measures.

We take notice the measure of ∠ C is missing. We can use the Triangle Angle Sum Theorem to find the missing measure. m ∠ C + m ∠ D + m ∠ E=180 ^(∘)Let's substitute the given values and solve the equation for m∠ K.

m∠ C+m∠ D+m∠ E=180^(∘)

SubstituteII

m∠ D= 46^(∘), m∠ E= 90^(∘)

m∠ C + 46^(∘)+ 90^(∘)=180^(∘)

AddTerms

Add terms

m∠ C+ 136^(∘)=180^(∘)

SubEqn

LHS-136^(∘)=RHS-136^(∘)

m∠ C=44^(∘)

We will add this information to our diagram.

Therefore, we see that ∠ C is the smallest angle, followed by ∠ D and then ∠ E. m ∠ CAngle-Side Relationships in Triangles.

Angle-Side Relationships in Triangles
If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle.

Now, considering this theorem, we will look for the corresponding opposite sides. cc Angle & Opposite Side [0.5em] ∠ C & DE [0.5em] ∠ D & CE [0.5em] ∠ E & CD Therefore, we can list the sides in order from smallest to largest. DE < CE < CD

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