--- title: "The quadrilateral family" subtitle: "Angles and shapes · Classifying shapes · Lesson 8" --- # The quadrilateral family **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** Is a square a rectangle? People argue about it, but mathematicians agree: yes. A name is a list of properties, and a square has every property on the rectangle's list. :::teacher-note **In one sentence:** Students drag quadrilaterals until the board names them, learn the defining properties of the trapezium, parallelogram, rhombus, rectangle, square and kite, and reason about the family (a square is a rectangle and a rhombus) (AC9M7SP02, node 2). **Students should be able to say:** I can name a quadrilateral from its properties. I can explain why a square is also a rectangle and a rhombus. I can use parallel sides to find angles in a parallelogram. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict What does a shape need to be a rectangle? :::keep-together **Is a square a rectangle?** ☐ No: a rectangle has two long sides and two short sides ☐ Only a big square ☐ Yes My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** Yes. A rectangle is a quadrilateral with four right angles, and a square has four right angles. It is a special rectangle, with its sides equal too. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## A family of properties :::keep-together ### Properties first Here is how to name a quadrilateral: check the properties, from the most special. 1. $\text{4 equal sides, 4 right angles} \Rightarrow \text{square}$ — Check the most special names first. 2. $\text{4 right angles} \Rightarrow \text{rectangle}$ — Right angles but sides not all equal. 3. $\text{4 equal sides} \Rightarrow \text{rhombus}$ — Equal sides but no right angles. 4. $\text{2 pairs parallel} \Rightarrow \text{parallelogram}$ — Then parallelogram, trapezium (at least one pair), kite. Now match each name to its property. **Match each quadrilateral to its defining property.** A. four equal sides B. two pairs of parallel sides C. two pairs of equal sides next to each other D. four right angles E. at least one pair of parallel sides {{columns 60% 40%}} {{rows 1}} | Part | Letter | |---|---| | `trapezium` | | | `parallelogram` | | | `rhombus` | | | `rectangle` | | | `kite` | | ::: :::teacher-note **Answer:** trapezium → E. at least one pair of parallel sides; parallelogram → B. two pairs of parallel sides; rhombus → A. four equal sides; rectangle → D. four right angles; kite → C. two pairs of equal sides next to each other A square has four equal sides and four right angles, so it fits the rhombus, rectangle and parallelogram rows too. ::: :::keep-together ### Find the mistake **Sam decides whether a square is a rectangle. Click the line where his reasoning goes wrong.** ``` 1 A square has four right angles. 2 A rectangle is a quadrilateral with four right angles. 3 A square also has two pairs of parallel sides, like a rectangle. 4 But a square is not a rectangle, because all its sides are equal. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 4 Line 4: having an extra property (equal sides) does not take away the rectangle's properties. A square has four right angles, so it is a rectangle: a special one. ::: :::keep-together ### Always true? **Which is always true?** ☐ Every kite is a rhombus ☐ Every rhombus is a parallelogram ☐ Every rectangle is a square ☐ Every parallelogram is a rhombus ::: :::teacher-note **Answer:** Every rhombus is a parallelogram A rhombus has two pairs of parallel sides, so it has the parallelogram's property: every rhombus is a parallelogram. ::: :::typical ## One change at a time **A rhombus.** Each task changes one thing. **A plain parallelogram.** **A kite.** **One right angle.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Naming quadrilaterals :::keep-together ### Most special name **A quadrilateral has two pairs of parallel sides, no right angles, and sides that are not all equal. What is its most special name?** ☐ Rectangle ☐ Trapezium ☐ Parallelogram ☐ Rhombus ::: :::teacher-note **Answer:** Parallelogram Two pairs of parallel sides, nothing more: a parallelogram. ::: :::keep-together ### All the names **Which names fit a square?** ☐ Square, rectangle, rhombus, parallelogram, trapezium and kite ☐ Square and rectangle only ☐ Square and rhombus only ☐ Only square ::: :::teacher-note **Answer:** Square, rectangle, rhombus, parallelogram, trapezium and kite A square has all these properties, so it belongs to all these families. ::: :::keep-together ### Angles in a parallelogram **One angle of a parallelogram is 70°. What is the angle next to it, in degrees?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 110 Neighbouring angles of a parallelogram are co-interior: 180 − 70 = 110°. ::: :::keep-together ### Review: triangles **Review: an isosceles triangle has two angles of 35°. What is the third, in degrees?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 110 180 − 35 − 35 = 110°. ::: ## Make: a right-angled trapezium Then: how many right angles can such a trapezium have, and why not three? Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **A shape has two pairs of parallel sides and four equal sides but no right angles. Give all its names.** {{lines-3}} ::: :::teacher-note **Answer:** Rhombus, parallelogram and kite. :::