# Using angle facts and logical reasoning to derive results

Students are required to know, understand and be able to apply angle facts appropriately. They are also required to use triangle congruence, similarity and properties of named quadrilaterals to derive results about angles and sides. Students also have to use transformational, axiomatic and property-based logical reasoning in order to solve problems.

Visit the secondary mathematics webpage to access all lists.

### Angles

This resource contains twenty two instant maths ideas which can be used as starters, extension questions or probing questions to assess understanding. Ideas include exploring the special names given to angles, the properties of parallel, perpendicular and skew lines, exploring the angles made by the hands of a clock at different times, proving that the angles on a line add up to 180 degrees and investigating which combinations of angles triangles can have.

### Angles

Students are required to know, and be familiar with, basic angle facts. **Angles** begins by explaining what an angle is and asks students to explore simple angles in terms of the points on a compass. The next section explains how to measure angles and is followed by a section requiring students to classify angles. There follows explanations, examples and exercises relating to angles on a line and angles at a point. This leads to a section in which students are asked to construct triangles, including questioning whether certain triangles can be constructed. The final section requires students to use the properties of triangles in order to find the size of the missing angles in triangles.

### Angle

This resource contains two packs of activities providing students with the opportunity to explore, investigate and practice angle facts within plane shapes. Appropriate activities in pack one include:

**Right-angle or not**: students identify right angles.

**Equal angles**: students identify angles of the same size.

**Logo is amazing**: students complete a variety of mazes using the program Logo.

Pack two contains:

**Radar** in which students explore polar coordinates

**Free hand angles**: students make different sized angles by folding circular pieces of paper.

Further activities applying angle properties include **Equiangular spirals** and **Patterns from spirals** and a number of activities relating to the use of bearings.

### Angle Properties

This resource contains many activities requiring students to use the properties of angles or shapes in order to complete the task or solve the problem. Appropriate tasks in pack one are: **Equal angles**: students need to understand that an angle is a measure of turn and **Logo is amazing**: students solve problems using the program Logo requiring a good sense of angle.

### GAIM Activities: Investigations

This resource contains a number of activities and investigations designed to use basic geometrical properties of shapes to investigate the situations presented or solve the problems posed. Some of the tasks may appear a little dated but the ideas are valid and can be used for inspiration when lesson planning.

Task 7, **Symmetry**, presents three shapes made from squares and asks students to make as many different symmetrical shapes as they can. This task can be followed by Task 8, **Polyominoes**, in which students are asked to make as many triominoes, tetrominoes and pentominoes as possible.

Task 14, **Triangles**, presents a 3 by 3 pin board and requires students to make as many different triangles as possible. Task 18, **Roofs** explores how to draw quadrilateral shaped roofs on isometric paper.

Task 21, **Crooked stars**, asks students to investigate tessellating shapes. Task 25 **Squares symmetry** requires students to investigate how many different symmetrical patterns can be found. Task 28, **Triangulating**, asks students to investigate how many triangles can be formed when diagonals of larger shapes are drawn.

Task 32 explores the similarities involved in international paper sizes. Task 35 requires students to solve problems based upon tetrominoes.

### Geometry and Logic

This resource from the Contemporary School Mathematics collection published by Edward Arnold, was written in the belief that there is still value in the examination of Euclidean geometry as a logical system, whatever other approach may be used to increase the students' spatial perception.

The topics include:**Chapter 6 page 59 Angles and circles:** This chapter includes terminology, angles in a segment, other angle properties of a circle, converses, formal proofs, tangents, alternate segment theorem, intersecting chords, the angle bisector theorem and exterior bisectors. This is a rigorous resource that could be used to challenge students aiming for the higher grades.

### Quad Tessellate

The initial challenge is to attempt to draw a quadrilateral that does not tessellate. There is a Geogebra file included with the resources that will help demonstrate the impossibility of the task. The Geogebra file works by updating a tessellation live as a vertex is dragged, giving a very pleasing visual representation of the result.

Having established in a visual sense that quadrilaterals tessellate, attention turns to providing a formal proof.