Question

The sum of the measures of the angles of a triangle is 180. The sum of the measures of the second and third angles is five times the measure of the first angle. The third angle is 16 more than the second. Let x, y, and z represent the measures of the first, second, and third angles, respectively. Find the measures of the three angles.

Answer 1

This question is solved by a system of equations. We have that:

• x is the first angle.
• y is the second angle.
• z is the third angle.

Doing this, we get that:

• The first angle measures 30º.
• The second angle measures 67º.
• The third angle measures 83º.

The sum of the measures of the angles of a triangle is 180.

This means that $x + y + z = 180$

The sum of the measures of the second and third angles is five times the measure of the first angle.

This means that:

$y + z = 5x$

From this, the first angle can be found:

$x + y + z = 180$

$x + 5x = 180$

$6x = 180$

$x = \frac{180}{6}$

$x = 30$

The measure of the first angle is of 30º.

The third angle is 16 more than the second.

This means that:

$z = 16 + y$

Since

$y + z = 5x$

$x = 30$

We get that the second angle is:

$y + 16 + y = 5(30)$

$2y + 16 = 150$

$2y = 134$

$y = \frac{134}{2}$

$y = 67$

The second angle measures 67º.

For the third angle:

$z = 16 + y = 16 + 67 = 83$

The third angle measures 83º.

A similar question is given at https://brainly.com/question/22826010

Answer 2

X measures 30°, Y measures 67°, and Z measures 83°.

Since the sum of the measures of the angles of a triangle is 180, and the sum of the measures of the second and third angles is five times the measure of the first angle, and the third angle is 16 more than the second, for find the measures of the three angles, the following calculation must be performed, posing linear equations:

• X + Y + Z = 180
• Y + Z = 5X
• Z = Y + 16

• X + 5X = 180
• 6X = 180
• X = 180/6
• X = 30

• Y + Y + 16 = 5 x 30
• 2Y + 16 = 150
• 2Y = 150 - 16
• Y = 134/2
• Y = 67

• Z = 67 + 16
• Z = 83

Therefore, X measures 30 °, Y measures 67 °, and Z measures 83 °.

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