Grade 8 Questions on Angles with Solutions and Explanations

Detailed solutions and full explanations to grade 8 math questions on angles are presented.

1. 
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   Solution

   The sum of all 3 interior angles of a triangle is equal to \(180^\circ\). Hence \[ 92^\circ + 27^\circ + x = 180^\circ \] Solve for \(x\) \[ x = 180^\circ - (92^\circ + 27^\circ) = 61^\circ \]
2. 
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   Solution

   Note that it is a right triangle. The sum of all 3 interior angles of the right triangle is equal to \(180^\circ\). Hence \[ y + 34^\circ + 90^\circ = 180^\circ \] Solve for \(y\) \[ y = 180^\circ - (90^\circ + 34^\circ) = 56^\circ \]
   
3. 
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   Solution

   Angle \(y\) and an angle of measure \(56^\circ\) are supplementary. Hence \[ y + 56^\circ = 180^\circ \] Solve for \(y\) \[ y = 180^\circ - 56^\circ = 124^\circ \] Angle \(x\) and an angle of measure \(144^\circ\) are supplementary. Hence \[ x + 144^\circ = 180^\circ \] Solve for \(x\) \[ x = 180^\circ - 144^\circ = 36^\circ \] The sum of angles \(x\), \(y\), and \(z\) of the triangle is equal to \(180^\circ\). \[ x + y + z = 180^\circ \] Substitute \(x\) and \(y\) by their values found above: \[ 36^\circ + 124^\circ + z = 180^\circ \] Solve for \(z\) \[ z = 20^\circ \]
   
4. 
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   Solution

   Let \(z\) be the third angle of the triangle on the right. The sum of the interior angles in the triangle on the right side is equal to \(180^\circ\). Hence \[ 26^\circ + 26^\circ + z = 180^\circ \] \[ z = 180^\circ - 26^\circ - 26^\circ = 128^\circ \] Angles \(z\) and \(y\) are supplementary. Hence \[ z + y = 180^\circ \] Solve for \(y\) \[ y = 180^\circ - z = 180^\circ - 128^\circ = 52^\circ \] The sum of the interior angles in the triangle on the left is equal to \(180^\circ\). Hence \[ x + y + 64^\circ = 180^\circ \] Solve for \(x\) \[ x = 180^\circ - 64^\circ - 52^\circ = 64^\circ \]
   
5. 
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   Solution

   Angle \(z\) and the angle of measure \(133^\circ\) are supplementary angles. Hence \[ z + 133^\circ = 180^\circ \] \[ z = 180^\circ - 133^\circ = 47^\circ \] The angles of the lower triangle add up to \(180^\circ\). Hence \[ 33^\circ + 133^\circ + x = 180^\circ \] \[ x = 180^\circ - 33^\circ - 133^\circ = 14^\circ \] The sum of the measures of the angles of the upper triangle is equal to \(180^\circ\). Hence \[ y + z + 114^\circ = 180^\circ \] \[ y = 180^\circ - 114^\circ - z, \quad z = 47^\circ \text{ (found before)} \] \[ y = 180^\circ - 114^\circ - 47^\circ = 19^\circ \]
   
6. 
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   Solution

   The sum of the angles of the triangle on the right side is equal to \(180^\circ\). Hence \[ w + 131^\circ + 32^\circ = 180^\circ \] \[ w = 180^\circ - 131^\circ - 32^\circ = 17^\circ \] Angle \(v\) and the angle with measure \(132^\circ\) are supplementary. Hence \[ 132^\circ + v = 180^\circ \] \[ v = 180^\circ - 132^\circ = 48^\circ \] The three angles of the triangle in the middle add up to \(180^\circ\). Hence \[ v + z + 122^\circ = 180^\circ \] \[ z = 180^\circ - 122^\circ - v \] \[ z = 180^\circ - 122^\circ - 48^\circ = 10^\circ, \quad v = 48^\circ \ \text{(found above)} \] Angle \(x\) and the angle with measure \(122^\circ\) are supplementary. Hence \[ x + 122^\circ = 180^\circ \] \[ x = 180^\circ - 122^\circ = 58^\circ \] The three angles of the triangle on the left side add up to \(180^\circ\). Hence \[ x + 43^\circ + y = 180^\circ \] \[ y = 180^\circ - 43^\circ - x \] \[ y = 180^\circ - 43^\circ - 58^\circ = 79^\circ, \quad x = 58^\circ \ \text{(found above)} \]

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