--- title: "Problems with quadrilaterals" subtitle: "Angles and shapes · Congruence and similarity · Lesson 19" --- # Problems with quadrilaterals **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** A gate is braced with a diagonal so it stays a rectangle. A carpenter checks it by measuring both diagonals. Why does that work, and what does it tell her? :::teacher-note **In one sentence:** Students use the angle and diagonal properties of quadrilaterals to find unknown angles and lengths, with reasons (AC9M8SP02, node 2). **Students should be able to say:** I can find unknown angles in parallelograms, rhombuses, kites and trapeziums. I can use diagonal properties to find lengths. I can give a reason for each step. **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 Here are the properties to use: | Shape | Angles | Diagonals | |---|---|---| | any quadrilateral | add to 360° | | | parallelogram | opposite angles equal; neighbours add to 180° | bisect each other | | rhombus | as a parallelogram | bisect each other at 90° | | rectangle | four right angles | equal, and bisect each other | | kite | the angles between the unequal sides are equal | cross at 90° | | isosceles trapezium | base angles equal (in pairs) | equal | Neighbouring angles of a parallelogram add to 180° because they are **co-interior** angles between parallel sides. :::keep-together **A parallelogram has one angle of 70°. What are the other three?** ☐ 110°, 110°, 70° ☐ 110°, 70°, 110° ☐ 70°, 70°, 70° ☐ You cannot tell My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 110°, 70° and 110°. The opposite angle is equal (70°), and each neighbour is 180° − 70° = 110°. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Find the angles :::keep-together ### Neighbours Here is the routine. 1. $\angle K = 72^\circ \text{ in parallelogram } KLMN$ — What is given? 2. $\angle K + \angle L = 180^\circ$ — Co-interior angles: KN is parallel to LM. 3. $\angle L = 180^\circ - 72^\circ = 108^\circ$ — Work it out. 4. $\angle M = 72^\circ,\ \angle N = 108^\circ$ — Opposite angles are equal. Check: they add to 360°. **In parallelogram PQRS, angle P = 65°. What is angle Q, in degrees?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 115 P and Q are co-interior angles between the parallel sides PS and QR: 180 − 65 = 115°. ::: :::keep-together ### Find the mistake **Jo works out the angles of a parallelogram. Click the line where she goes wrong.** ``` 1 ABCD is a parallelogram with angle A = 65°. 2 Opposite angles add to 180°, so angle C = 180° − 65° = 115°. 3 Angles A and B are co-interior, so angle B = 180° − 65° = 115°. 4 So the angles are A = 65°, B = 115°, C = 115° and D = 65°. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 2 Line 2: opposite angles are equal, so C = 65°. Neighbouring angles add to 180°. Then D = 115°. ::: :::keep-together ### A kite **Kite ABCD has AB = AD and CB = CD, and angle B = 100°. Which other angle must be 100°, and why?** ☐ Angle C: it is next to B ☐ Angle A: it is opposite ☐ Angle D: triangles ABC and ADC are congruent (SSS) ☐ None of them ::: :::teacher-note **Answer:** Angle D: triangles ABC and ADC are congruent (SSS) AB = AD, CB = CD and AC is shared, so the triangles are congruent and angle B = angle D. ::: :::typical ## Other shapes **Rhombus.** **Rectangle diagonals.** **Kite.** **Isosceles trapezium.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Quadrilateral problems :::keep-together ### Angles with x **Two neighbouring angles of a parallelogram are x° and 2x°. What is x?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 60 x + 2x = 180, so 3x = 180 and x = 60. The angles are 60° and 120°. ::: :::keep-together ### Rhombus sides **A rhombus has diagonals of 6 cm and 8 cm. How long is each side, in cm?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 5 The diagonals cut each other in half at 90°, making right-angled triangles with legs 3 and 4: the side is 5 cm. ::: :::keep-together ### Opposite angles **In parallelogram ABCD, angle A = 52°. What is angle C, in degrees?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 52 Opposite angles of a parallelogram are equal: 52°. ::: :::keep-together ### Review: similar shapes **Review: a 2 cm by 3 cm rectangle is enlarged with scale factor 4. What is the longer side now, in cm?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 12 3 × 4 = 12 cm. ::: ## Make: a right-angled kite Then read angle C on the board and work out angles B and D, with a reason for each. The board rounds each angle to a whole degree, so its B and D can differ by 1°: your answer can be a half. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **In parallelogram ABCD, angle B = 3 × angle A. Find all four angles.** {{lines-3}} ::: :::teacher-note **Answer:** A + 3A = 180, so A = 45°. B = 135°, C = 45°, D = 135°. :::