Missing angles of a triangle worksheet - Free Printable

I will solve this geometry problem by finding the missing angles in each of the three diagrams. I'll use fundamental geometric principles like the sum of angles in a triangle (180°), the properties of straight lines (180°), and vertical opposite angles.

Let me break it down section by section.

Part a)

This diagram shows a triangle with one exterior angle given as 125°.
- The exterior angle is equal to the sum of the two opposite interior angles.
- So, $125^\circ = 21^\circ + x$
- Therefore, $x = 125^\circ - 21^\circ = 104^\circ$

- Angle y is adjacent to the 125° angle on a straight line, so they are supplementary.
- $y + 125^\circ = 180^\circ$
- Therefore, $y = 180^\circ - 125^\circ = 55^\circ$

- The sum of all angles in a triangle is 180°.
- So, $21^\circ + x + z = 180^\circ$
- We already found $x = 104^\circ$, so:
- $21^\circ + 104^\circ + z = 180^\circ$
- $125^\circ + z = 180^\circ$
- Therefore, $z = 180^\circ - 125^\circ = 55^\circ$

Part b)

This diagram has intersecting lines and a triangle.
- Angles a and 113° are on a straight line, so they are supplementary.
- $a + 113^\circ = 180^\circ$
- Therefore, $a = 180^\circ - 113^\circ = 67^\circ$

- Angles b and 113° are vertical opposite angles, so they are equal.
- Therefore, $b = 113^\circ$

- In the triangle, we have angles 35°, c, and another angle that is vertically opposite to angle a (which we found is 67°). Since vertical opposite angles are equal, the third angle in the triangle is also 67°.
- Sum of angles in a triangle: $35^\circ + c + 67^\circ = 180^\circ$
- $102^\circ + c = 180^\circ$
- Therefore, $c = 180^\circ - 102^\circ = 78^\circ$

Part c)

This diagram shows two triangles sharing a vertex.
- Angles f and 34° are vertical opposite angles, so they are equal.
- Therefore, $f = 34^\circ$

- In the right triangle, the sum of angles is 180°.
- We have angles 34°, 55°, and m.
- $34^\circ + 55^\circ + m = 180^\circ$
- $89^\circ + m = 180^\circ$
- Therefore, $m = 180^\circ - 89^\circ = 91^\circ$

- Angle p is adjacent to the 55° angle on a straight line, so they are supplementary.
- $p + 55^\circ = 180^\circ$
- Therefore, $p = 180^\circ - 55^\circ = 125^\circ$

- In the left triangle, we have angles 71°, g, and n.
- Also, angles n and f are vertical opposite angles, so $n = f = 34^\circ$.
- Sum of angles in the triangle: $71^\circ + g + n = 180^\circ$
- $71^\circ + g + 34^\circ = 180^\circ$
- $105^\circ + g = 180^\circ$
- Therefore, $g = 180^\circ - 105^\circ = 75^\circ$

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Final Answers:

a)
- $x = 104^\circ$
- $y = 55^\circ$
- $z = 55^\circ$

b)
- $a = 67^\circ$
- $b = 113^\circ$
- $c = 78^\circ$

c)
- $f = 34^\circ$
- $g = 75^\circ$
- $m = 91^\circ$
- $n = 34^\circ$
- $p = 125^\circ$