McGraw Hill Integrated II, 2012
MH
McGraw Hill Integrated II, 2012 View details
3. Inequalities in One Triangle
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Exercise 34 Page 431

Find m∠ MSR using the fact that ∠ VSR and ∠ MSR form a linear pair.

SM< MR

Practice makes perfect

In order to determine the relationship between SM and MR, we can use Theorem 5.10 about Angle-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.

With that being understood, let's now take a closer look at the given diagram.
We can see that ∠ MSR is opposite to MR and ∠ MRS is opposite to SM. However, the diagram does not show a measure for ∠ MSR. To find this measure, notice that ∠ VSM and ∠ MSR form a linear pair. The sum of their measures is 180^(∘). m∠ VSM + m∠ MSR =180^(∘) From the diagram, we know that ∠ VSM measures 110^(∘). Let's substitute this value into this equation and calculate the measure of ∠ MSR.
m∠ VSM + m∠ MSR =180^(∘)
110^(∘) + m∠ MSR =180^(∘)
m∠ MSR =70^(∘)
We can add this piece of information to the diagram.

Because ∠ MSR is greater than ∠ MRS, we know that MR is longer than SM. SM< MR