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Question 1
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Triangles ABC and XYZ are congruent, and the measure of angle A = 5x + 3, the measure of angle B = 3x + 23, and the measure of angle X = 7x - 27. Find the value of x.

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x =

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Question 2
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In an isosceles triangle, the vertex angle is 28 degrees. What are the measures of each of the base angles?(1 point)
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1 answer

To solve the problems you've presented:

Question 1:

Triangles ABC and XYZ are congruent, which means corresponding angles are equal. We have:

  • Measure of angle A = \(5x + 3\)
  • Measure of angle B = \(3x + 23\)
  • Measure of angle X = \(7x - 27\)

Since angles A and X are corresponding angles in the congruent triangles, we set them equal to each other:

\[ 5x + 3 = 7x - 27 \]

Now solve for \(x\):

  1. Rearranging the equation: \[ 5x + 3 - 7x + 27 = 0 \] \[ -2x + 30 = 0 \] \[ -2x = -30 \] \[ x = 15 \]

Now we have found that:

\( x = 15 \)

Question 2:

In an isosceles triangle, the vertex angle is 28 degrees. Let's denote the base angles as \(B\). Since the triangle's angles sum up to 180 degrees, we can write the equation:

\[ 28 + 2B = 180 \]

Now solve for \(B\):

  1. Subtract 28 from both sides: \[ 2B = 180 - 28 \] \[ 2B = 152 \]

  2. Divide by 2: \[ B = 76 \]

So, the measures of each of the base angles are:

\( 76 \) degrees.

Final answers:

  • Question 1: 15
  • Question 2: 76