Section A — Chapters 1–22

Geometry II: Theorems

Junior Cycle — 1st Year

✓By the end of this lesson students will be able to understand what an axiom is in geometry.
✓By the end of this lesson students will be able to identify and calculate vertically opposite angles.
✓By the end of this lesson students will be able to identify and calculate alternate and corresponding angles when parallel lines are intersected by a transversal.
✓By the end of this lesson students will be able to apply the angle properties of triangles (sum of angles, exterior angle) to solve problems.

Key concepts

Axiom

An axiom is a statement that is accepted as true without proof. It is a fundamental truth in geometry that we use as a starting point for reasoning. For example, 'Through any two distinct points there is exactly one line' is an axiom.

Vertically Opposite Angles

When two straight lines intersect, they form four angles. The angles that are directly opposite each other at the point of intersection are called vertically opposite angles. Vertically opposite angles are always equal.

Parallel Lines and Transversals

Parallel lines are lines that are always the same distance apart and never meet. We use arrows on the lines to show they are parallel (e.g., AB || CD). A transversal line is a line that intersects two or more other lines. When a transversal intersects two parallel lines, special angle relationships are formed.

Corresponding Angles

When a transversal intersects two parallel lines, corresponding angles are in the same relative position at each intersection. They often form an 'F' shape. Corresponding angles are equal.

Alternate Angles

When a transversal intersects two parallel lines, alternate angles are on opposite sides of the transversal and between the two parallel lines. They often form a 'Z' shape. Alternate angles are equal.

Angles in a Triangle

A triangle is a three-sided polygon. The sum of the three interior angles of any triangle is always 180°.

∠A + ∠B + ∠C = 180°

Exterior Angle of a Triangle

An exterior angle of a triangle is formed when one side of the triangle is extended. The exterior angle is equal to the sum of the two opposite interior angles.

Exterior Angle = Sum of two opposite interior angles

Key facts to remember

1An axiom is a statement accepted as true without proof.
2Vertically opposite angles are equal.
3When two parallel lines are cut by a transversal, corresponding angles are equal.
4When two parallel lines are cut by a transversal, alternate angles are equal.
5The sum of the three interior angles in any triangle is 180°.
6An exterior angle of a triangle is equal to the sum of the two opposite interior angles.
7Angles on a straight line add up to 180°.

Worked examples

Example 1

In the diagram, two straight lines intersect. Find the values of angles x, y, and z.

IAngle x and the angle 70° are vertically opposite angles.

IITherefore, x = 70° (Vertically opposite angles are equal).

IIIAngle y and the angle 70° are angles on a straight line.

IVTherefore, y + 70° = 180° (Angles on a straight line).

Vy = 180° - 70°

VIy = 110°

VIIAngle z and angle y are vertically opposite angles.

VIIITherefore, z = y (Vertically opposite angles are equal).

9z = 110°

Answer

x = 70°, y = 110°, z = 110°

Remember that angles on a straight line add up to 180°.

Example 2

In the diagram, line AB is parallel to line CD (AB || CD). The transversal line EF intersects them. Find the values of angles a and b.

IAngle a and the angle 65° are corresponding angles (F-shape).

IISince AB || CD, corresponding angles are equal.

IIITherefore, a = 65°.

IVAngle b and the angle 65° are alternate angles (Z-shape).

VSince AB || CD, alternate angles are equal.

VITherefore, b = 65°.

Answer

a = 65°, b = 65°

Always look for the 'F' shape for corresponding angles and the 'Z' shape for alternate angles when parallel lines are involved.

Example 3

In triangle PQR, ∠P = 50° and ∠Q = 75°. Find the value of ∠R. Also, if side QR is extended to S, find the exterior angle ∠PRS.

IThe sum of the interior angles in a triangle is 180°.

IISo, ∠P + ∠Q + ∠R = 180°.

III50° + 75° + ∠R = 180°.

IV125° + ∠R = 180°.

V∠R = 180° - 125°.

VI∠R = 55°.

VIIThe exterior angle ∠PRS is equal to the sum of the two opposite interior angles (∠P and ∠Q).

VIII∠PRS = ∠P + ∠Q.

9∠PRS = 50° + 75°.

10∠PRS = 125°.

Answer

∠R = 55°, ∠PRS = 125°

You could also find ∠PRS by noting that ∠R and ∠PRS are angles on a straight line, so ∠PRS = 180° - ∠R = 180° - 55° = 125°.

Common mistakes

✗Assuming lines are parallel when it is not stated or indicated by arrows on the lines.
✗Confusing corresponding angles with alternate angles, or vice versa.
✗Incorrectly applying the exterior angle theorem, for example, adding an adjacent interior angle instead of the two opposite ones.
✗Not stating the geometric reason for each step in an angle calculation, which is often required for full marks.

Exam tips

★Always draw diagrams clearly and label all known angles and any angles you need to find.
★When working with parallel lines, look for the 'F' shape for corresponding angles and the 'Z' shape for alternate angles.
★For every step in your calculation, state the geometric reason (e.g., 'vertically opposite angles', 'angles in a triangle'). This shows your understanding.
★After finding your answers, quickly check if they make sense. For example, do angles on a straight line add to 180°? Do angles in a triangle add to 180°?

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