Grade 8 Questions on Angles with Solutions and Explanations

This page is designed to help students, parents, and teachers master finding unknown angles in triangles through practice questions and step-by-step solutions. Expand the hidden solutions beneath each question to reveal the reasoning behind every step.

To successfully solve these problems, remember the following key geometry rules:

Practice Questions

  1. Find the unknown angle \(x\): Triangle with one unknown angle and two known angles (92 and 27 degrees)
    View Step-by-Step Solution

    The sum of all 3 interior angles of a triangle is equal to \(180^\circ\).

    • \(92^\circ + 27^\circ + x = 180^\circ\) (Set up the equation)
    • \(x = 180^\circ - (92^\circ + 27^\circ)\) (Isolate \(x\))
    • \(x = 180^\circ - 119^\circ\)
    • \(x = 61^\circ\)
  2. Find the unknown angle \(y\): Right triangle with one 34 degree angle and one unknown angle
    View Step-by-Step Solution

    Note that the square symbol indicates a right triangle. The sum of all 3 interior angles of the right triangle is equal to \(180^\circ\).

    • \(y + 34^\circ + 90^\circ = 180^\circ\) (Set up the equation using the \(90^\circ\) angle)
    • \(y = 180^\circ - (90^\circ + 34^\circ)\) (Isolate \(y\))
    • \(y = 180^\circ - 124^\circ\)
    • \(y = 56^\circ\)
  3. Find the unknown angles \(x\), \(y\), and \(z\): Triangle with three unknown interior angles and given supplementary exterior angles
    View Step-by-Step Solution

    Step 1: Find \(y\)

    • Angle \(y\) and the external angle of \(56^\circ\) form a straight line, so they are supplementary.
    • \(y + 56^\circ = 180^\circ\)
    • \(y = 180^\circ - 56^\circ\) = \(124^\circ\)

    Step 2: Find \(x\)

    • Angle \(x\) and the external angle of \(144^\circ\) are supplementary.
    • \(x + 144^\circ = 180^\circ\)
    • \(x = 180^\circ - 144^\circ\) = \(36^\circ\)

    Step 3: Find \(z\)

    • The sum of the interior angles \(x\), \(y\), and \(z\) is \(180^\circ\).
    • \(x + y + z = 180^\circ\)
    • \(36^\circ + 124^\circ + z = 180^\circ\) (Substitute the known values)
    • \(160^\circ + z = 180^\circ\)
    • \(z = 180^\circ - 160^\circ\) = \(20^\circ\)
  4. Find the unknown angles \(x\) and \(y\): Two adjacent triangles sharing a side, with multiple unknown angles
    View Step-by-Step Solution

    Step 1: Find the missing angle in the triangle on the right

    • Let \(z\) be the third top angle of the triangle on the right. The sum of its interior angles is \(180^\circ\).
    • \(26^\circ + 26^\circ + z = 180^\circ\)
    • \(z = 180^\circ - 52^\circ\) = \(128^\circ\)

    Step 2: Find \(y\)

    • Angles \(z\) and \(y\) sit on a straight line, making them supplementary.
    • \(z + y = 180^\circ\)
    • \(128^\circ + y = 180^\circ\)
    • \(y = 180^\circ - 128^\circ\) = \(52^\circ\)

    Step 3: Find \(x\)

    • The sum of the interior angles in the triangle on the left is \(180^\circ\).
    • \(x + y + 64^\circ = 180^\circ\)
    • \(x + 52^\circ + 64^\circ = 180^\circ\) (Substitute \(y\))
    • \(x + 116^\circ = 180^\circ\)
    • \(x = 180^\circ - 116^\circ\) = \(64^\circ\)
  5. Find the unknown angles \(x\), \(y\), and \(z\): Two vertically stacked triangles with three unknown angles
    View Step-by-Step Solution

    Step 1: Find \(z\)

    • Angle \(z\) and the angle measuring \(133^\circ\) are supplementary (on a straight line).
    • \(z + 133^\circ = 180^\circ\)
    • \(z = 180^\circ - 133^\circ\) = \(47^\circ\)

    Step 2: Find \(x\)

    • The angles of the lower triangle add up to \(180^\circ\).
    • \(33^\circ + 133^\circ + x = 180^\circ\)
    • \(166^\circ + x = 180^\circ\)
    • \(x = 180^\circ - 166^\circ\) = \(14^\circ\)

    Step 3: Find \(y\)

    • The sum of the measures of the angles of the upper triangle is equal to \(180^\circ\).
    • \(y + z + 114^\circ = 180^\circ\)
    • \(y + 47^\circ + 114^\circ = 180^\circ\) (Substitute \(z\))
    • \(y + 161^\circ = 180^\circ\)
    • \(y = 180^\circ - 161^\circ\) = \(19^\circ\)
  6. Find the unknown angles \(v\), \(w\), \(x\), \(y\), and \(z\): Three connected triangles with five unknown angles
    View Step-by-Step Solution

    Step 1: Find \(w\)

    • The sum of the angles of the triangle on the right side is \(180^\circ\).
    • \(w + 131^\circ + 32^\circ = 180^\circ\)
    • \(w + 163^\circ = 180^\circ\)
    • \(w = 180^\circ - 163^\circ\) = \(17^\circ\)

    Step 2: Find \(v\)

    • Angle \(v\) and the angle with measure \(132^\circ\) are supplementary.
    • \(132^\circ + v = 180^\circ\)
    • \(v = 180^\circ - 132^\circ\) = \(48^\circ\)

    Step 3: Find \(z\)

    • The three angles of the middle triangle add up to \(180^\circ\).
    • \(v + z + 122^\circ = 180^\circ\)
    • \(48^\circ + z + 122^\circ = 180^\circ\) (Substitute \(v\))
    • \(170^\circ + z = 180^\circ\)
    • \(z = 180^\circ - 170^\circ\) = \(10^\circ\)

    Step 4: Find \(x\)

    • Angle \(x\) and the angle measuring \(122^\circ\) are supplementary.
    • \(x + 122^\circ = 180^\circ\)
    • \(x = 180^\circ - 122^\circ\) = \(58^\circ\)

    Step 5: Find \(y\)

    • The three angles of the triangle on the left side add up to \(180^\circ\).
    • \(x + 43^\circ + y = 180^\circ\)
    • \(58^\circ + 43^\circ + y = 180^\circ\) (Substitute \(x\))
    • \(101^\circ + y = 180^\circ\)
    • \(y = 180^\circ - 101^\circ\) = \(79^\circ\)

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