# Space, shape and measurement: Explore, interpret and justify geometric relationships

### Subject outcome

Subject outcome 3.2: Explore, interpret and justify geometric relationships

### Learning outcomes

- Use geometry of straight lines and triangles to solve problems and to justify relationships in geometric figures. Concepts to include are:
- angles of a triangle
- exterior angles
- straight lines
- vertically opposite angles
- corresponding angles
- co-interior angles
- alternate angles.

- State and apply the following theorems of circles:
- If a line is drawn from the centre of a circle to the midpoint of a chord, then that line is perpendicular to the chord.
- If a line is drawn from the centre of the circle perpendicular to the chord, then it bisects the chord.
- If an arc subtends an angle at the centre of the circle and at any point on the circumference, then the angle at the centre is twice the measure of the angle at the circumference.
- If the diameter of a circle subtends an angle at the circumference, then the angle subtended is a right-angle triangle.
- If an angle subtended by a chord at a point on the circumference is a right angle, then the chord is a diameter.
- Angles in the same segment of a circle are equal.
- The opposite angles of a cyclic quadrilateral are supplementary.
- An exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.
- If the exterior angle of a quadrilateral is equal to the interior opposite angle the quadrilateral will be a cyclic quadrilateral.
- The four vertices of a quadrilateral in which the opposite angles are supplementary will be a cyclic quadrilateral.
- If a tangent to a circle is drawn, then it is perpendicular to the radius at the point of contact.
- If a line is drawn perpendicular to a radius at the point where the radius meets the circle, then it is a tangent to the circle.
- If two tangents are drawn from the same point outside a circle then they are equal in length.
- The angle between a tangent to a circle and a chord drawn from the point of contact is equal to an angle in the alternate segment (tan-chord theorem).

**Note:**Proofs of the above theorems are excluded

### Unit 1 outcomes

By the end of this unit you will be able to:

- Use properties of angles on straight lines and angles in a triangle to find unknown angles including:
- angles of a triangle
- exterior angles
- straight lines
- vertically opposite angles
- corresponding angles
- co-interior angles
- alternate angles

### Unit 2 outcomes

By the end of this unit you will be able to:

- Apply the following theorems relating to circles:
- Line drawn perpendicular to a chord from the centre of the circle bisects the chord, and its converse (line drawn from the circle centre to the mid-point of chord is perpendicular to the chord).
- Angle at the centre of a circle is twice the size of the angle at the circumference.
- Diameter of a circle subtends a right angle at the circumference, and its converse (if an angle subtended by a chord at a point on the circumference is a right angle, then the chord is a diameter).
- Angles in the same segment of a circle are equal.

### Unit 3 outcomes

By the end of this unit you will be able to:

- Define a cyclic quadrilateral.
- Apply the theorem opposite angles of a cyclic quadrilateral are supplementary.
- Apply the theorem exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.
- Apply the converses of equality between opposite angles, between angles in the same segment, and between exterior angles and interior opposite angles, to prove a quadrilateral is cyclic.

### Unit 4 outcomes

By the end of this unit you will be able to:

- Apply the theorem tangent perpendicular to radius at point of contact.
- Apply the converse theorem of tangent perpendicular to radius.
- Apply the theorem two tangents drawn from the same point outside a circle are equal.
- Apply the tangent-chord theorem.