For the given triangles, we will find the range of possible values for x. Let's start by labeling the vertices of the triangles.
In order to do that, we will use the Hinge Theorem to compare AB and CD. Therefore, we should find the measure of ∠ CAD. Notice that the sum of m∠ ACB and m∠ ACD is 90^(∘).
60+m∠ ACD=90 ⇒ m∠ ACD=30
From here, by the Triangle Angle Sum Theorem, we can find m∠ CAD.
All x-values greater than 7 will work. Besides that m∠ CAD is greater than m ∠ DCA. By the Triangle Larger Angle Theorem, the side opposite the larger angle is longer than the side opposite the lesser angle.
3x+17 > 5x
Let's solve this inequality for x.
Therefore, 8.5 is the upper limit of x. We can determine the range of possible values of x by combining the two inequalities.
Inequality I:&& 7 < & x
Inequality II:&& & x< 8.5
Combined:&& 7<& x<8.5
