--- title: "Proving quadrilateral properties" subtitle: "Angles and shapes · Congruence and similarity · Lesson 18" --- # Proving quadrilateral properties **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** You measure the diagonals of one rectangle: both 10 cm. Is that true for every rectangle in the world? Measuring can't check them all, but a proof can. :::teacher-note **In one sentence:** Students test diagonal properties of quadrilaterals on the board, then prove them with congruent triangles (AC9M8SP02, node 1). **Students should be able to say:** I can find two triangles in a quadrilateral and say which test makes them congruent. I can use matching parts of congruent triangles to prove a property. I can explain why measuring one shape is not a proof. **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 :::keep-together **Ana draws one rectangle and measures both diagonals: 10 cm each. Has she proved that every rectangle's diagonals are equal?** ☐ Yes, if she draws two more ☐ Yes: she measured carefully ☐ No: one example is not a proof My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** No. Examples suggest the rule, but a proof gives reasons that work for every rectangle, such as two congruent triangles. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Write the proof :::keep-together ### The proof Here is a proof that a parallelogram's diagonals cut each other in half. Every proof in this lesson has the same plan. 1. $\text{Parallelogram } ABCD,\ \text{diagonals cross at } M$ — Look at triangles ABM and CDM. 2. $AB = CD$ — Opposite sides of a parallelogram are equal. 3. $\angle BAM = \angle DCM,\ \angle ABM = \angle CDM$ — Alternate angles: AB is parallel to DC. 4. $\triangle ABM \equiv \triangle CDM\ (\text{ASA})$ — Name the test. 5. $AM = CM,\ BM = DM$ — Matching sides are equal: the diagonals cut each other in half. **Put the proof that a rectangle's diagonals are equal in order.** A. `Two sides and the angle between them match, so △ABC ≡ △BAD (SAS).` B. `Side AB is in both, BC = AD (opposite sides of a rectangle), and angle B = angle A = 90°.` C. `The diagonals look the same length, so they are equal.` D. `Matching sides of congruent triangles are equal, so AC = BD: the diagonals are equal.` E. `ABCD is a rectangle. Look at triangles ABC and BAD, made by the diagonals AC and BD.` Correct order (letters): {{field-24}} ::: :::teacher-note **Answer:** see the program below Find two triangles, give the equal parts with reasons, name the test, then use matching sides. ```maths ABCD is a rectangle. Look at triangles ABC and BAD, made by the diagonals AC and BD. Side AB is in both, BC = AD (opposite sides of a rectangle), and angle B = angle A = 90°. Two sides and the angle between them match, so △ABC ≡ △BAD (SAS). Matching sides of congruent triangles are equal, so AC = BD: the diagonals are equal. ``` ::: :::keep-together ### Find the mistake **Leo tries to prove a property of kites. Click the line where his reasoning goes wrong.** ``` 1 Leo wants to prove that a kite's diagonals cross at 90°. 2 He draws one kite carefully and measures where the diagonals cross. 3 It measures 90°, so this proves it for every kite. 4 He writes: the diagonals of a kite are perpendicular. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 3 Line 3: one measured kite is an example, not a proof. A proof gives reasons that hold for every kite. ::: :::keep-together ### A kite **Kite ABCD has AB = AD and CB = CD. The diagonal AC splits it into triangles ABC and ADC. Which test shows they are congruent?** ☐ They are not congruent ☐ SAS ☐ SSS: AB = AD, CB = CD and AC is shared ☐ RHS ::: :::teacher-note **Answer:** SSS: AB = AD, CB = CD and AC is shared AC is in both triangles, so all three sides match: SSS. Then angle BAC = angle DAC, so AC cuts angle A in half. ::: :::typical ## More properties **A kite.** **A rhombus.** **A parallelogram.** **A square.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Reasons :::keep-together ### Parallelogram **In parallelogram ABCD, the diagonal AC makes triangles ABC and CDA. AB = CD and BC = DA (opposite sides). Which test shows they are congruent?** ☐ SSA ☐ They are not congruent ☐ RHS ☐ SSS: AC is shared ::: :::teacher-note **Answer:** SSS: AC is shared AB = CD, BC = DA and AC is in both: SSS. So angle B = angle D: opposite angles are equal. ::: :::keep-together ### Rhombus **Which is a proof that a rhombus's diagonals cross at 90°?** ☐ I measured one and got 90° ☐ The four small triangles are congruent by SSS, so the four angles at the centre are equal and add to 360° ☐ All rhombuses are squares ☐ It looks like a plus sign ::: :::teacher-note **Answer:** The four small triangles are congruent by SSS, so the four angles at the centre are equal and add to 360° Four equal angles that make 360° are 90° each. ::: :::keep-together ### Which shape **Which quadrilateral's diagonals always cut each other in half, but are not always equal?** ☐ A rectangle ☐ A square ☐ A kite ☐ A parallelogram ::: :::teacher-note **Answer:** A parallelogram Every parallelogram's diagonals bisect each other; they are equal only when it is a rectangle. ::: :::keep-together ### Review: parallel lines **Review: two parallel lines are cut by a transversal. One co-interior angle is 115°. What is the other, in degrees?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 65 Co-interior angles add to 180°: 180 − 115 = 65°. ::: ## Make: equal, not halved Then write a proof (or the plan of one) that its diagonals are equal. Which two triangles would you use? Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Kite ABCD has AB = AD and CB = CD. Prove that angle B = angle D.** {{lines-3}} ::: :::teacher-note **Answer:** Triangles ABC and ADC: AB = AD, CB = CD, AC shared, so they are congruent (SSS). Angle B and angle D match, so they are equal. :::