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Math  /  Geometry

QuestionFind the range of possible values for xx. \qquad
The range is □\square <x<<x< □\square (Simplify your answers.)

Studdy Solution

STEP 1

1. The triangle is isosceles, which means two sides are equal.
2. The two equal sides are each 9.
3. The base angles are (x−3)∘(x-3)^\circ and 27∘27^\circ.
4. The sum of angles in a triangle is 180∘180^\circ.
5. We need to find the range of possible values for xx.

STEP 2

1. Determine the third angle of the triangle.
2. Set up the equation for the sum of angles in the triangle.
3. Solve for xx and find the range of possible values.

STEP 3

Since the triangle is isosceles, the base angles are equal. Therefore, (x−3)∘(x-3)^\circ must be equal to 27∘27^\circ.

STEP 4

Set up the equation for the sum of angles in the triangle:
(x−3)+27+third angle=180 (x-3) + 27 + \text{third angle} = 180
Since (x−3)∘(x-3)^\circ and 27∘27^\circ are the base angles, the third angle is also (x−3)∘(x-3)^\circ.

STEP 5

Substitute the third angle with (x−3)∘(x-3)^\circ in the equation:
(x−3)+27+(x−3)=180 (x-3) + 27 + (x-3) = 180

STEP 6

Simplify and solve for xx:
2(x−3)+27=180 2(x-3) + 27 = 180 2x−6+27=180 2x - 6 + 27 = 180 2x+21=180 2x + 21 = 180 2x=180−21 2x = 180 - 21 2x=159 2x = 159 x=1592 x = \frac{159}{2} x=79.5 x = 79.5
The range of possible values for xx is:
79.5<x<80 \boxed{79.5 < x < 80}

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