Question

1. suppose the green thumb club decided that angle 2 (east corner) must be 35 for it to fit into an existing design. if angle 2 were 35, what would the new value of x be? what would the measures of angle 1 and angle 3 be in this new scenario? (assume the expressions remain the same).

x=

new angle 1=

new angle 2=

explain if these new angles would form a valid triangle and why or why not.

Explanation:

Step1: Find the value of \(x\)

The sum of angles in a triangle is \(180^{\circ}\). Given one angle is \(35^{\circ}\), let the other two angles be \(x\) and \(35^{\circ}\). So, \(x + 35+35=180\).

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Step2: Determine new angles

New Angle 1: If we split the \(110^{\circ}\) angle. Let's assume a valid split (for example, if we split it into two angles such that the sum of all three angles in the new triangle is \(180^{\circ}\)). Suppose we have angles \(a\), \(b\), and \(35^{\circ}\) where \(a + b=110\). Then \(a + b+35=(a + b)+35=110 + 35=145
eq180\) (this is wrong approach, actually using triangle angle - sum property correctly)

Wait, no. Wait, original triangle had angles \(35^{\circ}\), \(35^{\circ}\), \(110^{\circ}\). If we change \(x = 110\) (say we make a new triangle). Let's use the triangle angle - sum formula \(A + B + C=180\). Suppose New Angle 1 \(=m\) and New Angle 2 \(=n\). Then \(m + n+35 = 180\), so \(m + n=145\). As long as \(m>0\), \(n>0\) and \(m + n = 145\), the new angles \(m\), \(n\), \(35^{\circ}\) will form a valid triangle. For example, if \(m = 50\) and \(n = 95\), \(50+95 + 35=180\)

Answer:

\(x = 110\). New Angle 1 and New Angle 2 (as long as their sum is \(145^{\circ}\)) will form a valid triangle because the sum of angles in a triangle is \(180^{\circ}\). If we call the two new angles \(A\) and \(B\) such that \(A + B=145^{\circ}\), then \(A + B+35^{\circ}=(A + B)+35^{\circ}=145^{\circ}+35^{\circ}=180^{\circ}\) which satisfies the triangle angle - sum property.